In my original answer (under this new answer) I mistakenly understood the lost sheep to be a righteous remnant of the northern kingdom. I have learned that instead they were "dead bones" and had to first be gathered by Jesus and his disciples and then be resurrected on Pentecost and up to the return of Christ in 70ad.
Israel started out with a man named Jacob who was renamed by God to Israel. He had twelve sons and these became known as "the twelve tribes of the house [IE: "family"] of Israel".
Later, Israel split into two houses, the northern house of Israel and the southern house of Judah. Over time the house of Israel abandoned YHVH and was rejected, to no longer be God's special people. Essentially they became gentiles, forgot their identity and were "lost to history". Or so it seemed. But God promised that while the lion's share of the northern kingdom would never be recovered a chosen number of their descendants would return to God and become sons of God. These, along with the remnant from Judea constitute the 144,000 of Revelation 7.
[Jer 16:13-21 ASV] (13) therefore will I cast you [the northern kingdom of Israel] forth out of this land into the land that ye have not known, neither ye nor your fathers; and there shall ye serve other gods day and night; for I will show you no favor. (14) Therefore, behold, the days come, saith Jehovah, that it shall no more be said, As Jehovah liveth, that brought up the children of Israel out of the land of Egypt; (15) but, As Jehovah liveth, that brought up the children of Israel from the land of the north, and from all the countries whither he had driven them. And I will bring them again into their land that I gave unto their fathers. (16) Behold, I will send for many fishers, saith Jehovah, and they shall fish them up; and afterward I will send for many hunters, and they shall hunt them from every mountain, and from every hill, and out of the clefts of the rocks. (17) For mine eyes are upon all their ways; they are not hid from my face, neither is their iniquity concealed from mine eyes. (18) And first I will recompense their iniquity and their sin double, because they have polluted my land with the carcasses of their detestable things, and have filled mine inheritance with their abominations. (19) O Jehovah, my strength, and my stronghold, and my refuge in the day of affliction, unto thee shall the nations come from the ends of the earth, and shall say, Our fathers have inherited nought but lies, even vanity and things wherein there is no profit. (20) Shall a man make unto himself gods, which yet are no gods? (21) Therefore, behold, I will cause them to know, this once will I cause them to know my hand and my might; and they shall know that my name is Jehovah.
God predicted through Ezekiel 37 that first he would gather and assemble the dead bones of Israel and afterwords breathe into the assemblage and resurrect them as a powerful army:
[Eze 37:5-14 ASV] (5) Thus saith the Lord Jehovah unto these bones: Behold, I will cause breath to enter into you, and ye shall live. (6) And I will lay sinews upon you, and will bring up flesh upon you, and cover you with skin, and put breath in you, and ye shall live; and ye shall know that I am Jehovah. (7) So I prophesied as I was commanded: and as I prophesied, there was a noise, and, behold, an earthquake; and the bones came together, bone to its bone. (8) And I beheld, and, lo, there were sinews upon them, and flesh came up, and skin covered them above; but there was no breath in them. (9) Then said he unto me, Prophesy unto the wind, prophesy, son of man, and say to the wind, Thus saith the Lord Jehovah: Come from the four winds, O breath, and breathe upon these slain, that they may live. (10) So I prophesied as he commanded me, and the breath came into them, and they lived, and stood up upon their feet, an exceeding great army. (11) Then he said unto me, Son of man, these bones are the whole house of Israel: behold, they say, Our bones are dried up, and our hope is lost; we are clean cut off. (12) Therefore prophesy, and say unto them, Thus saith the Lord Jehovah: Behold, I will open your graves, and cause you to come up out of your graves, O my people; and I will bring you into the land of Israel. (13) And ye shall know that I am Jehovah, when I have opened your graves, and caused you to come up out of your graves, O my people. (14) And I will put my Spirit in you, and ye shall live, and I will place you in your own land: and ye shall know that I, Jehovah, have spoken it and performed it, saith Jehovah.
So Jesus and his disciples went "fishing for men" and assembled the bones and after his death, resurrection and ascension poured out holy breath on the bones and they "stood up as a mighty army".
Hosea also spoke of this and Paul quotes Hosea:
[Hos 1:6-11 ASV] (6) And she conceived again, and bare a daughter. And Jehovah said unto him, Call her name Lo-ruhamah; for I will no more have mercy upon the house of Israel, that I should in any wise pardon them. (7) But I will have mercy upon the house of Judah, and will save them by Jehovah their God, and will not save them by bow, nor by sword, nor by battle, by horses, nor by horsemen. (8) Now when she had weaned Lo-ruhamah, she conceived, and bare a son. (9) And Jehovah said, Call his name Lo-ammi; for ye are not my people, and I will not be your God. (10) Yet the number of the children of Israel shall be as the sand of the sea, which cannot be measured nor numbered; and it shall come to pass that, in the place where it was said unto them, Ye are not my people, it shall be said unto them, Ye are the sons of the living God. (11) And the children of Judah and the children of Israel shall be gathered together, and they shall appoint themselves one head, and shall go up from the land; for great shall be the day of Jezreel.
[Rom 9:25-29 ASV] (25) As he saith also in Hosea, I will call that my people, which was not my people; And her beloved, that was not beloved. (26) And it shall be, that in the place where it was said unto them, Ye are not my people, There shall they be called sons of the living God. (27) And Isaiah crieth concerning Israel, If the number of the children of Israel be as the sand of the sea, it is the remnant that shall be saved: (28) for the Lord will execute his word upon the earth, finishing it and cutting it short. (29) And, as Isaiah hath said before, Except the Lord of Sabaoth had left us a seed, We had become as Sodom, and had been made like unto Gomorrah.
Though Jesus was not sent to gather in gentiles (he would send Paul to do that later) he did not despise the Canaanite woman who reached out to him for help.
For an excellent explanation of the background of the lost sheep of the house of Israel please see this video. You can skip the 3 minute intro. Also I do not agree with the second half.
While the two previous answers (JBH and SolaGatia) are valid I would want to point out a few things that will not only help understand this passage but the scriptures in general.
First of all, the "lost sheep of the house of Israel" are not "unsaved Jews" but rather the faithful Jews who had no faithful shepherds. They were part of Yehovah's devoted "sheep" but scattered and troubled because the leadership of the Jews did not care for them as they ought to have:
NIV Ezekiel 34: 1The word of the Lord came to me: 2“Son of man, prophesy against the shepherds of Israel; prophesy and say to them: ‘This is what the Sovereign Lord says: Woe to you shepherds of Israel who only take care of yourselves! Should not shepherds take care of the flock? 3You eat the curds, clothe yourselves with the wool and slaughter the choice animals, but you do not take care of the flock. 4You have not strengthened the weak or healed the sick or bound up the injured. You have not brought back the strays or searched for the lost. You have ruled them harshly and brutally. 5So they were scattered because there was no shepherd, and when they were scattered they became food for all the wild animals. 6My sheep wandered over all the mountains and on every high hill. They were scattered over the whole earth, and no one searched or looked for them.
7“ ‘Therefore, you shepherds, hear the word of the Lord: 8As surely as I live, declares the Sovereign Lord, because my flock lacks a shepherd and so has been plundered and has become food for all the wild animals, and because my shepherds did not search for my flock but cared for themselves rather than for my flock, 9therefore, you shepherds, hear the word of the Lord: 10This is what the Sovereign Lord says: I am against the shepherds and will hold them accountable for my flock. I will remove them from tending the flock so that the shepherds can no longer feed themselves. I will rescue my flock from their mouths, and it will no longer be food for them.
11“ ‘For this is what the Sovereign Lord says: I myself will search for my sheep and look after them. 12As a shepherd looks after his scattered flock when he is with them, so will I look after my sheep. I will rescue them from all the places where they were scattered on a day of clouds and darkness. 13I will bring them out from the nations and gather them from the countries, and I will bring them into their own land. I will pasture them on the mountains of Israel, in the ravines and in all the settlements in the land. 14I will tend them in a good pasture, and the mountain heights of Israel will be their grazing land. There they will lie down in good grazing land, and there they will feed in a rich pasture on the mountains of Israel. 15I myself will tend my sheep and have them lie down, declares the Sovereign Lord. 16I will search for the lost and bring back the strays. I will bind up the injured and strengthen the weak, but the sleek and the strong I will destroy. I will shepherd the flock with justice. ... 22I will save my flock, and they will no longer be plundered. I will judge between one sheep and another. 23I will place over them one shepherd, my servant David, and he will tend them; he will tend them and be their shepherd. 24I the Lord will be their God, and my servant David will be prince among them. I the Lord have spoken. ... 30Then they will know that I, the Lord their God, am with them and that they, the Israelites, are my people, declares the Sovereign Lord. 31You are my sheep, the sheep of my pasture, and I am your God, declares the Sovereign Lord.’ ”
So this mission of Jesus was to locate the faithful but scattered Jewish "sheep" of Yehovah God and to tend to them. By saying "I am THE good shepherd" Jesus claims that he is the "David" prophesied in Ezekiel:
NIV John 10:
11“I am the good shepherd [promised to the Jews]. The good shepherd lays down his life for the sheep. 12The hired hand is not the shepherd and does not own the sheep. So when he sees the wolf coming, he abandons the sheep and runs away. Then the wolf attacks the flock and scatters it. 13The man runs away because he is a hired hand and cares nothing for the sheep.
The "other sheep" are the gentiles:
NIV John 10: 14“I am the good shepherd; I know my sheep and my sheep know me— 15just as the Father knows me and I know the Father—and I lay down my life for the sheep. 16 I have other sheep that are not of this sheep pen. I must bring them also. They too will listen to my voice, and there shall be one flock and one shepherd. 17The reason my Father loves me is that I lay down my life—only to take it up again. 18No one takes it from me, but I lay it down of my own accord. I have authority to lay it down and authority to take it up again. This command I received from my Father.”
So the sheep that loved Yehovah God also responded to Jesus because they recognized that he was speaking what God had taught him:
BSB John 14: 10Do you not believe that I am in the Father and the Father is in Me? The words I say to you, I do not speak on My own. Instead, it is the Father dwelling in Me, carrying out His work.
So while Jesus was on this mission specifically to gather the lost sheep of Israel under Paul's ministry there would be an establishment of a new assembly where the distinction of Jew vs gentile would not limit the flock (Eph 2). This woman, by her faith received some of what was then promised only to the Jews.