Paul emphasizes the importance of prayer in 1st Timothy 2. I believe this emphasis is to be applied in both our private praying as well as in public worship. Prayer is to be a priority according to verse 1: “I urge, then, first of all, that requests, prayers, intercession and thanksgiving be made for everyone.”
As the church gathers there should be a time for bringing personal requests before God in behalf of everyone. Since the context of this paragraph has to do with reaching the world with the good news these requests would be not only for believers who have needs but also non-believers. Intercession here has an emphasis on coming into intimate relationship with God as you pray.
These prayers are to include requests for political leaders and others in positions of authority. This would include those in authority in the church.
Prayer is a privilege for which we should be thankful, and as we pray, we should be thankful for the anticipated outcome. The picture of tranquility in our social setting, and modeling a life of godliness and holiness, sets the stage for sharing the good news.
All of this is in keeping with God’s desire to see all men saved and therefore pleases Him. It also makes the presentation of the gospel clearer. If people are living in turmoil, salvation for them might be to just get delivered from the stress. If being terrorized by authorities, it might suggest deliverance from this tyranny. But in an atmosphere of peace and quietness the heart of the gospel can be explained as the need to be reconciled to God through Christ, our mediator, by faith.