Washtenaw County's cities all(?) have an elected Mayor.

Both Ann Arbor and Ypsilanti have the weak mayor (council/manager) form of government. This means that the Mayor basically functions as an at-large Councilmember and presides over City Council meetings, while a hired, professional City Manager handles day-to-day operations of the City. This means that, while the Mayor has a lot of symbolic importance as the voice and face of the City, and a major role in shaping the city's policy agenda, no, the Mayor cannot "fix" your parking tickets. (By contrast, "strong mayor" cities, like Detroit and Flint, put the Mayor's office in direct control over significant portions of city administration.)