What was the relationship between local government and the Puritan churches?
In the early days the British colonies of New England, the Puritan churches and the local governments were essentially one and the same thing. There was no separation of church and state. Instead, the government was made up of church leaders and enforced church law.
This can be seen in some of the words and actions of people from the Massachusetts Bay Colony. This was the colony led by Governor John Winthrop. He is the one that said the colony shold become a "city upon a hill," showing the world how a truly Christian society would be run. In colonies such as this, men who were not part of the Puritan church did not have the right to vote. The colonies could and did throw out people who did not believe the proper things. The most famous examples of this were the expulsions of Anne Hutchinson and Roger Williams from the Bay Colony in the 1630s.
In short, then, there was no real difference between the Puritan churches and the local governments in the early days of colonization in New England.