I would argue that the voters have the most impact on the executive branch in a municipal government that runs on a strong mayor system. There is much more impact in this sort of system than there is in a weak mayor system or, even more, in a council-manager system.
In a strong mayor system, the voters get to vote directly for a mayor who has a great deal of power over the executive branch. This gives the voters fairly direct impact on that branch. By contrast, in other systems of government, the executive branch is run by unelected officials like a city manager. These forms of government do not give the people nearly as much direct impact on the executive branch.