Have sb doing sth = 使得某人…… (to directly provoke a reaction)
have (someone) eating out of the palm of (one’s) hand
To have great influence over someone.
If you want to have someone eating out of the palm of your hand, flattery always helps.
Have sb do sth = 吩咐某人……(to pay/ask sb to do sth)
I had the gardener dig up those horrible weeds.