There isn't any single answer to that question! I'm not sure what angle you're approaching from - what is it that needs to be dealt with?
If you're a fellow councillor, maybe you need to consider whether any apology is needed from the mayor or from the council - it would probably be better to act as a council rather than as an individual. And if this is a serious problem, perhaps the council should make a different choice next May when the council year starts and a new chairman is selected.
If you're a member of the public, then you could ask council members to consider what they might do to deal with any problems arising out of the mayor's behaviour. In the end, it's down to the electorate to put forward suitable candidates and vote for them.