What tensions exist between national and state governments on policy matters, and what unique powers do mayors and governors enjoy?

National and state government have different and overlapping set of laws and jurisdictions,

Explanation:

National Government of US does not have complete right over the states in that the states are able to decide their won laws and penal codes until they are not in direct conflict with the Union which is never looked for and thus never exposed.

The governor of a state an have absolute control over the state's penal code apart from the country and the mayor of a single city is able to control the work of the police in that place.