I wonder how Mamdani will handle these situations. Does he have the authority to tell Times Square to shut down their power consumption?

surely the city (not the mayor alone) can define non-essential power consumption and say “hey cut that shit out or get fined”?

I am assuming the city owns the power company. if not, wtf lol

They do not own the power company, no. ConEd is private, but the state has some authority over it.
He does have the authority to send whoever operates it a letter asking them to pretty please shut it down. Unlike with regular people, the chance the recommendation will be followed is zero.