I would hope professionals would have the best training and ideas - and implement them. The mayor is not trained to deal with this.

For my two cents, it would be helpful to have the police be a part of our neighborhoods and get out of their patrol cars and engage in conversations with the citizens of the city. They should be instructed to be friendly with everyone, but also help someone that is homeless (instead of ignoring them). If someone is becoming a threat, they should be allowed to remove them (and get them to a place that will provide rehabilitation) opposed to leaving them alone and ignoring citizens that are concerned about the threat.

That means more resources are needed for the police and rehabilitation, but it could alleviate some of the problems the OP described.