Rom. 1.9-10: Paul prays for a change in his circumstances. It doesn't come about.
Rom. 8.26-27: We are to live by the Spirit, not in sin. We must depend on God, and the Holy Spirit helps us in our weakness, even when we don't know what to pray for. Then why should we bother to pray at all, since our prayers are from a position of weakness, and we sometimes find ourselves without words? It’s because even in that situation our prayers can get through to the Lord because of the Spirit’s intercession. We don’t need to be strong or a Shakespeare. Just pray.
Rom. 10.1: Pray for spiritual transformation in the hearts of people.
Rom. 15:30-32: Paul prayed to be delivered from the unbelievers in Judea, but that didn't happen. He was arrested and sent to Rome where he spent years in jail.
2 Cor. 1.11: Paul had been through the most terrible of circumstances. God didn't change any of those for him. Paul reports to us that they didn't die, thanks to the prayers of the Corinthian people. What did the prayers accomplish? They spared Paul from death, but didn't seem to reduce his life-threatening experiences.
2 Cor. 12.8: Paul prayed for a change of circumstances, but that didn't happen.
Eph. 1.15-19a: Paul prays for spiritual and internal things: wisdom, revelation, enlightenment, and power.
Eph. 3.14-19: Everything Paul prays for is spiritual and internal. The power of God is at work, not in our circumstances, but "within us."
Eph. 6.18-20: Nothing is off-limits to our prayers. Pray about everything. But when Paul asks them to pray for him, he asks that words will be fed to him when he speaks for the Lord about the mysteries of salvation. He asks also that he will be protected from fear. Both of these requests pertain to his inner self, his mind and his soul.
Phil. 1.4-6, 9-11: He prays about their internal and spiritual state.
Phil. 1.19: Paul believed in the effectiveness of prayer. The Philippians pray for a changed of circumstance for him—his deliverance, and Paul believed that their prayers would accomplish this change. It is unknown to us if this prayer was answered.
Phil 4.6: Pray about anything and everything, both generally and by making requests. We are invited to share our hearts with God, to present our anxieties, and to seek his peace. Our prayers cover every single aspect of our lives.
Phil. 4.19: Paul can't repay the Philippians, but he prays that God will. It could be material in nature, but is not necessarily so.
Col. 1.3-4; 1 Thes. 1.2-3: Prayer is an opportunity to show our gratitude to God. And what does he mention? Faith, love, and that the gospel is growing and bearing fruit.
Col. 1.9-12: Paul prays for a world of spiritual benefits.
Col. 4.3: Is he praying for circumstances, or for people hearts? it's hard to tell. Probably both.
1 Thes. 3.10-13: Paul prays about a circumstance which we don't know if it came about or not. Other than that, he prays for spiritual benefits: love, strength, holiness.
2 Thes. 1.11-12: Despite all the circumstantial trouble (1.4), Paul prays that they would behave worthy of their calling, and that god would bless the acts of their faith, and in the process bring glory to Jesus.
2 Thes. 3.1, 2: Verse 1 is about spiritual matters. Verse 2 is about circumstances: protection and deliverance. It's possible this prayer was answered (Acts 18.1-17).
1 Tim. 2.1-2, 8: Paul prays about salvation for people. but he also requests prayer for government leaders. We have to be convinced that God has the capacity to intervene in the minds and lives of all government officials. We have a responsibility for those in government. And the reason we pray for our government officials? Godliness and holiness. Again, the focus seems to be largely on internal, spiritual characteristics and values and only slightly on anything circumstantial, though prayer about circumstances is certainly encouraged.
1 Tim. 4.5: Prayer is a means of consecration. But pray for others. Never stop praying for others.
1 Tim. 5.5: Paul prays that needs will be met. This is clearly circumstances of her life.
2 Tim. 1.3: Paul is praying, but about what? In the subsequent verses he brings up faith, spiritual gifts, power, love, self-discipline, courage to testify, suffering, and a holy life. If these are what was on Paul’s mind, his prayers were mostly for their inner beings and their spiritual strength, not so much about their circumstances.
Philemon 1.4: Here we see prayers of thanksgiving for the faith and love Paul sees in Philemon. He is not asking for favors in circumstances, but expressing his gratitude for the inner spiritual character traits he sees in his friend.
Philemon 1.6: Paul prays about Philemon's spiritual maturity.
Philemon 1.22: Paul prays about his circumstances. it is unknown whether or not this prayer of request was ever answered.
Conclusion?
Paul is clear: pray about EVERYTHING. Leave no stone unturned. But his example and his life show that prayer is mostly all about our inner lives: our souls, peace, strength, wisdom, joy, growth, and salvation. It is by our prayers that lives our changed: our own, and those of others. Prayer is a means of consecration, and an opportunity to express thankfulness.