Can an elected politician appoint someone to do their job?

For the president the first answer that comes to mind is the cabinet. What if as the newly elected mayor of a small town I looked into myself and saw that I was not the best person in the world, but thought hey I could find someone and pay them handsomely to run this city. If that person fails…I’ll just fire them. And find someone new.

Hmmm…...what then…