Depending where you are in South Manchester Carrington riding school has a good reputation for children, Daisy Nook and Croft end in Tameside are also good or Hargate Hill in Glossop. I’m originally from that neck of the woods and my little niece rides at Daisy Nook.
Carrington is fine - NHOH had a few early lessons there before I let him sit on one of mine. Very easy location from Sale and would recommend over some of the other local options.