Thank you everyone for your replies I have read, some of them several times and think I understand it all!!!
I probably should have said that she has been riding at the riding school for 6 months, but has been riding a friends horse, on the lunge, walking and trotting, since she was much younger, so all in all about 20 months. We have had the pony since June and this week is the first time she has cantered her and it has only been twice. I was impressed she could manage to circle half the school and go large again to be honest, I was only mastering the trot at her age (by watching other people,no formal lessons for me).
The pony isn't unsafe and has never 'run off' as such, only tried to nap to the gate, especially if going past it, but dd is able to see this and react now, so that is much better as she has become a more active rider.
Marialuisa - I think the pony is opening her mouth (I presume why the instructor tightened the noseband), and she is just strong. She does come back, but races.
CMOT - have done lots on transitions and working on her seat with sitting trot with/without stirrups, now she is cantering will be working on those transistions too, but because up to now she has been so steady in trot, have not done fast trot/walk transitions, what a good idea!! Duh!
JRs - You describe what is happening better than me, the pony is running and I am sure it is because dd is erratic. Her seat is actually very good, I was surprised how solid she is in the saddle, but her hands do come up. I am meeting the owner on Thursday morning, so will discuss it with her, at least I feel a bit more informed now.
Saggy - I do agree with you. The pony is so sweet, kind and gentle, I don't want to spoil her, we didn't actually know she was so forward going until we started to canter and that isn't a bad thing really. DD will learn in time and I too am always asking 'do you want to ride tonight', I just am so proud of how gutsy and keen she is, I can't believe she rides like she does to be honest. I do try not to be pushy. Luckily she loves it, she told me that when she was in the lesson, before the scary bit, she felt 'Whee!!' and wanted to shout out 'I love riding', really loud, but didn't because of the new instructor.
Trebles - Yes dd is inexperienced, but I really don't think she is overhorsed, no I don't want to truss the pony up, I used to hate seeing kids at shows with every gaget going on their ponies. I do want to try and find some solution, as there has to be the cross over from where you are on ploddy ponies that are totally dead to the leg to a more forward going pony that you can progress on.
Thank you once again for all your replies. It has helped [she says off to Google some of the more unusual, obscure bits mentioned above]