1)The MTA is a state agency under the governor and not the mayor
2) The state has had special powers over the budget of the city since the state bailed out the city financially during the 70s
I’m not aware of any special powers still in effect; it would be good to have a link for that.
Title IX of the NYS Constitution describes the general terms of home rule, including the rules that govern all counties, towns, and cities within the state. I can’t find an explicit reference anywhere to NYC or its countries anywhere within that title, so I don’t know which section you think implies “special powers.”
(If anything, NYC is afforded an unusual degree of home rule relative to other municipalities in NY, as evidenced by the fact that it's really 5 counties in a trenchcoat.)