The Mayor of London doesn't have any power over the City of London Corporation. They are completely separate authorities.

The Corporation is essentially a unitary/borough-tier local authority, overseeing the "square mile" centre of the city, and has a council of elected councilmen. It provides housing, education, social services, street cleaning, markets etc for a small area of central London, and has existed since time immemorial.

The Mayor's remit, which has only existed since the year 2000, covers the whole 600-square mile area of Greater London, and provides strategic services like transport, strategic planning, fire and rescue, and the metropolitan police.

The Mayor of London wouldn't have had any involvement in this at all.

Fun fact, the City of London is the last local authority in the UK where businesses as well residents get to vote. Businesses can appoint one voter for every five employees up to 50, and then one per 50 employees after that.

Not to be confused with the Lord Mayor of London.