Philippians is a 'prison epistle', as we can see from verse 1:7, where Paul speaks of being in bonds - or imprisoned:
Philippians 1:7 (KJV): Even as it is meet for me to think this of you all, because I have you in my heart; inasmuch as both in my bonds, and in the defence and confirmation of the gospel, ye all are partakers of my grace.
In verse 19 Paul says the deliverance from his present predicament will surely come about from their prayers and the Spirit of Jesus Christ. In verses 20-25, Paul considers the possibility of execution, although (v20) his hope - and his expectation - is of release. He is uncertain (vv 23-24) whether it is better to depart this life and be with Christ, or to stay and be of service to the Philippians. In verse 25, Paul repeats his confidence that he will be released:
Philippians 1:19-25: For I know that this shall turn to my salvation through your prayer, and the supply of the Spirit of Jesus Christ, According to my earnest expectation and my hope, that in nothing I shall be ashamed, but that with all boldness, as always, so now also Christ shall be magnified in my body, whether it be by life, or by death. For to me to live is Christ, and to die is gain. But if I live in the flesh, this is the fruit of my labour: yet what I shall choose I wot not. For I am in a strait betwixt two, having a desire to depart, and to be with Christ; which is far better: Nevertheless to abide in the flesh is more needful for you. And having this confidence, I know that I shall abide and continue with you all for your furtherance and joy of faith;
A Trinitarian view would be that "πνεύματος Ἰησοῦ Χριστοῦ" is a reference to the Holy Spirit, but another possibility is that Paul thought of πνεύματος as the life-giving breath of Jesus. The Cambridge Bible for Schools and Colleges considers the alternatives and concludes that the reference must be to the Holy Spirit (Paraclete). Cambridge then says the phrase may mean either “the supply which is the Spirit” or “the supply which the Spirit gives.” The two practically converge, but Cambridge prefers the former meaning.