To have somebody do something = to get somebody to do something; a one-off action, something that’s not supposed to take very long.
To have somebody doing something = to cause someone to be occupied in doing something, usually a continuous activity; something that is going to be a long process.
Jane had Frank book her tickets for the show. (Jane got Frank to book her tickets for the show).
He had the gardener take care of his lawn.
I’ll have another gardener looking after these camellia bushes that are planted in the flower bed by the front porch.
The management will have another coach working (work) with the team in the next session.
Both expressions may sound similar, but to have someone do something is more appropriate for somebody giving orders, though to have someone doing something can also have this sense:
He had me letting (let) out small snores when mother came into the room.
Cousin Tom had me ignoring Sarah and Nancy all through these days.
We also have the construction to have + something (object) + done (past participle) by somebody to express the same:
Jane had her tickets booked by the agent. (Jane got the agent to book her tickets).
Notice that the agent (‘by somebody’) is often omitted:
Did you have your car wash (by the attendant)?
Did you get your car wash (by the attendant)?
Some more examples:
They should have someone check it out for them.
She can’t believe he did this to her! He had her doubt her fashion sense!
Mrs Joseph will have miss Sharon explain the midpoint formula to her niece. (a one-off action, something that’s going to take little time).
Mrs Joseph will have miss Sharon teaching her niece. (a more continuous action)
Some grammar experts say the second structure is not acceptable and even ungrammatical and the first one is correct. Anyway, both constructions are in use. Only problem with the second is that it is somewhat ambiguous and we should stick to the first one.
Some more examples:
She has the maid clean the house. (the maid cleans the house regularly for her)
She will have the maid cleaning the house. (the maid will clean the house in near future)
She is having the maid clean the house. (the maid will clean the house in near future)
He is going to have the actors of the movie photograph with us after the shooting.
Don’t be stupid! I’m not going to have anybody laugh at you.
She was a couple of hours late. It had me wondering if there was something wrong.