First of all, then, I urge that supplications, prayers, intercessions, and thanksgivings be made for all people, for kings and all who are in high positions, that we may lead a peaceful and quiet life, godly and dignified in every way. (1 Tim. 2:1-2)
In the church's corporate worship, we ought to pray for various concerns — for the local church's own need, for the needs of the broader church, for the lost, and for the good of society and the government. It is with regards to the latter that the issue of confusion of Church and State might arise. After all, the Church is to be in the world but not of the world. How are we to pray for any country without trying to co-opt God for earthly and sometimes nationalistic agendas? Both sides in the American Civil War thought God was on their side, and the Battle Hymn of the Republic is a "patriotic hymn" for such an occasion. In the American context, how should one pray for the country without trying to sound like the "Republican or Democratic party at prayer"? Should one pray for military success and thus the "successful bombing and killing" of America's enemies as if they were God's enemies?
It is here that I think the 2-Kingdoms theory and the spirituality of the church helps. The Church's concerns are to be heavenly, not earthly. Furthermore, God is not American. God does not treat America differently from any other nation, or Singapore for that matter - all equally insignificant, all equally dispensable. All countries exist as God wishes, and cease to exist once He decided they have finished their part to play in history. God does not care about "human rights" so called; He does whatever He pleases, regardless of what autonomous Man thinks or schemes. If God decides to destroy America or Singapore tomorrow, nothing anyone can do will alter the fact — that's how insignificant we humans are.
The Church in any country is always a heavenly creature. While her members are citizens of various countries, the Church should not be tied to any country. The Church's obligation to pray for leaders and authorities does not change regardless of what government or what conqueror (in times of war) is in power, and therefore we do not adopt the particular agendas of countries, nations and states.
Prayers for authorities focuses thus on requests for freedom and absence of persecution from authorities, for general human flourishing, and for the government to uphold God's moral law, which anyone from any country can pray without taking sides on issues of real-politics. One does not have to be a citizen of a country to pray that the government would uphold the moral law of God, for all countries everywhere ought to do the same. The moral law stands over all nations, and God will punish those who violate His law, nations as well as individuals. The country that celebrates wickedness will be judged, and God does NOT respect the "rights" of those celebrating sin.
The kings of the earth set themselves, and the rulers take counsel together,
against the LORD and against his Anointed, saying,
“Let us burst their bonds apart and cast away their cords from us.”
He who sits in the heavens laughs; the Lord holds them in derision. (Ps. 2:2-4)
2-Kingdoms theory allows both citizens and non-citizens together to pray for those in authority. A foreigner can pray for the good of the host nation, for Christians ought not to be nationalistic and seek only the good of their own countries. We ought to pray for the government to have the fear of God and uphold the moral law of God, to rule justly and fairly for the good of all, especially for the people God has temporarily placed under them, which is our civic duty. And through all these, may the peace that comes from governments that uphold justice serve the mission of the church, so that many will be reached for Christ and His Gospel.