The powers of a mayor of a town or city are Veto Power.
The Veto Power that a mayor has is the power to unilaterally stop an official action.
For example a mayor can hire and fire city or town employees and oversees the operations of the town or city.
The mayor also decides on if something needs to be done and figures out if the town or city can afford to have the work done.
Mayors also have the power to work on improvements within the city or town.
For example if enough people want to get a new park in town and everyone votes on it then the mayor can then decide to go through with it or not.
The mayor can also write in city ordinances after a vote goes through with the city council.
The mayor can then also enforce those city ordinances and take people into city or town court for those violations of those city ordinances.
Basically a mayor of a town is like the president of the United States.
But the mayor is just a president of the city or town.