What part does God play in Hobbes' politics?
In answering this, we should remember that Hobbes's main concern was that the sovereign should be all-powerful. He felt that an absolute ruler was the only sort of ruler who could ensure that people would not fall back into the war of each against all. This informs Hobbes's thinking about God and religion in politics.
Hobbes wanted complete religious tolerance so long as the ideas espoused by the religion did not undermine the sovereign. He believed that religious conflict was a major source of instability in society and he therefore wanted to prevent such conflict. If all people were completely free to believe as they wished, religious conflict would be minimized. However, this also meant that religious action (as opposed to belief) had to be controlled by the sovereign. People should be free to believe, but not free to act as they wished. The sovereign should have the right to prohibit any sort of acts based on religious belief that would undermine social stability.
Hobbes's view of religion and God, then, was strongly impacted by his belief that a ruler had to have absolute power. He did not want religious belief to infringe on that power.