The government is an extension of her majesty, and since the Queen can't even enter without prior permission.... though I believe there's some form of power exerted on the City by parliament.
They are governed by the "City of London Corporation" which is much different from the "Mayor of London".
They have their own police force, power stations, elections. Etc;
Unless you have information I don't have.
Specifically, the City of London Corporation is headed by the Lord Mayor of the City of London, who is a completely different person than the Mayor of London who heads the Greater London Authority, which is the government of what most people think of as the city named London, which is a completely different thing than the City of London.
I mean, Manchester has its own police force, Merseyside has its own police, and so forth. Each borough in the England has its own elections. What's peculiar about the City is that the electorate of the City's elections includes businesses as well as residents. But it's not some kind of extraterritorial oddity as some conspiracy theorists think.
There's a commons MP for the City of London _and_ Westminster (jointly) but that arrangement is... peculiar and not like other regions of the UK at all. She doesn't officially represent the CoL from their perspective.
There _is_ a member of the House of Lords who is directly posted by CoL and his job is to prevent the erosion of rights that City of Londoners enjoy.
[0] https://www.cityoflondon.gov.uk/about-us/about-the-city-of-l...
This was because the Congress had been beseiged by disgruntled citizens when it met in Philadelphia, and the state of Pennsylvania had refused to intervene.
I'm getting this from Wikipedia: https://en.wikipedia.org/wiki/Washington,_D.C.#History