This definition of ON should help
7c—used as a function word to indicate the focus of obligation or responsibility
drinks are on the house
put the blame on me
Without more context of what the article is discussing, I can't explain what it means in detail. But I'm quite sure it means that the mayor has the responsibility of the city, and the city is (if figuratively) doing very poorly or (if literally) burning from riots.
You may have heard a simple phrase using "on" this way:
This is on you!
Meaning: "This was your responsibility!" and usually "This is your fault!"
So the article is placing the blame and responsibility on the Mayor for what is happening in the city.