What business has Professor Bhaer been taking care of while in town?

He's a tutor in New York to support his orphaned half-american nephews.

Explanation:

Professor Bhaer is a character from Little Women. He became a controversial character because of the romance generalisations that tend to appear in romantic novels.

The fact that he's non-attractive, old and a bit silly, make readers not like him at the beginning. Even-though, he wins Jo's love, because of the respect she showed to him, his education with a classical tone and his unique accent that attracts all attentions.

Bhaer is a middle-aged German professor who WORKS AS A TUTOR in New York in order to support his orphaned, half-american nephews. Jo meets him because she is working at the bording house. They fall in love. Professor Bhaer marries Jo and after the marriage they open their own bording school for boys.