An excellent discussion of this question can be found in “The Synoptic Problem – Four Views” (hereafter “Four Views”), edited by Porter and Dyer. (See esp. pp. 39-40, 54-56, 80-82, 119-125)
Andreas Ennulat has suggested that there are more than 1,000 places in which Matthew & Luke agree against Mark (Die “Minor Agreements”: Untersuchungen zu einer offenen Frage des synoptischen Problems) in either common addition or common omission.
The minor agreements – or not-so-minor-agreements as some would label them – are frequently used as evidence against the Two Source Hypothesis. Both the Farrer Hypothesis and the Two-Gospel Hypothesis suggest that the minor agreements demonstrate that Luke used Matthew as a source, thereby dispensing with the need to appeal to the hypothetical source Q.
Defenders of the Two Source Hypothesis have suggested a variety of possibilities to account for this evidence, including scribal harmonization, the influence of oral tradition, and multiple editions of the Gospels (Four Views p. 124). Allowing for multiple versions of each Gospel opens up a wide variety of considerations—if you can imagine it you can build a theory around it—but like the Q document this possibility suffers from a complete absence of evidence in the manuscripts or patristic writings.
Scribal edits at this level would suggest a significant early effort to harmonize the Gospels, but when in the early Christian movement did a centralized structure exist that could both coordinate the effort and disseminate it? (and disseminate it so thoroughly that all evidence of other readings has been lost). By the time the church developed its 4th century structure, it was far too late to manufacture the minor agreements, as we have Gospel manuscripts that predate this period. (e.g. P4, P45, P64, P75, and others. A nice summary of early manuscripts with links to greater to detail is available on Wikipedia: https://en.wikipedia.org/wiki/List_of_New_Testament_papyri)
Oral tradition may be a promising avenue. It would have to be a very tightly controlled oral tradition though. How was it controlled?
Additionally, the possibility of Mark-Q overlap is sometimes suggested to account for Matthew-Luke agreements, although there is a risk in doing so: the number of agreements is so great that Q starts looking more and more like the Gospel of Matthew, defeating the idea of a Q document in the first place. (see One Gospel from Two - Mark’s Use of Matthew and Luke, pp. 6-7)
In speaking of the minor agreements Craig Evans, a proponent of the Two Source Hypothesis, has acknowledged “Supporters of the Two Source Hypothesis agree that this is the most vulnerable point in their hypothesis.” (Four Views p. 40)
The Farrer Hypothesis suggests that the best explanation is that Matthew, writing second, used Mark as a source, and Luke, writing third, used both. The Two Gospel Hypothesis suggests that the best explanation is that Luke wrote second using Matthew as a/the principle source. David Barrett Peabody has written “If the hundreds of ‘minor agreements’ scattered throughout the Triple Tradition are seen to merge with and form a pattern with the scores of ‘major agreements’…then it becomes increasingly apparent that Luke was primarily dependent on Matthew and not Mark, and there is no need for Q or the priority of Mark.” (One Gospel from Two - Mark’s Use of Matthew and Luke, p. 6)
The following thought experiment is interesting. I give a story to two students, tell them to work separately, and ask them to re-write the story, add their own touch to it, and make it nearly twice as long. Later, the students come back with their revisions—and although there are numerous differences, they have also made exactly the same change in the same place in the story not once or twice, but hundreds of times! Am I going to appeal to chance or to collusion? Perhaps I’ll give them the benefit of the doubt and search for a similar story that both of them might have read, to provide such common inspiration. But if none can be found, I’m going to strongly suspect collaboration or collusion.
In my view the minor agreements of Matthew & Luke against Mark are devastating evidence against the popular Two Source Hypothesis, and efforts to harmonize the theory with the minor agreements—while plausible—come up of short on the evidence. The cleanest solution common in the literature—requiring no hypotheticals—is that Luke knew Matthew’s Gospel.