Ask security to check the cameras to see if they are leaving early or showing up late. Ask them about login times to see if people are signing in or out early.
Ask whomever tracks the cars to see if they are being signed in or out at odd times. Even asking them just for a list of times a specific car was signed in or out.
At both of these points there is a person who decides whether the mayor should be allowed specific information given their duties. (In the OP example it's for a time sheet that they manage). Also note, both of these provide less information than the OP.