1.
to humor someone; to try to please; to curry favor with
To act in a way designed to make someone happy or to get on their good side, often by flattering them or doing what they want. Can have a slightly negative connotation of being insincere.
機嫌 means mood or humor, and とる here means to take or handle. The expression describes actively trying to manage or improve someone else's mood.
The expression can be neutral (comforting a child) or slightly negative (being obsequious to a superior). Context determines the nuance.