There's two ways of looking at this:
I think I've talked more parents out of buying horses for their kids that anyone in the world and instead, I'd advise lessons, lessons, and more lessons.
1. You should call your trainer back to evaluate if at all possible. Sometimes parents think that after a handful of lessons that they are ok to then go off on their own prematurely and then get into a bit of trouble. Not saying this is the case with your daughter and her trainer, I don't know. You've only had Philly about a month or so and I do hate to see you give up on her so soon. That's barely enough time for Philly to settle in. Philly may also resent having to have a variety of riders. Sometimes horses that have been used as school horses get to hate having to put up with different mounts after a while. Perhaps you can get the trainer back for a while and she can get her bitted up properly and schooled on the lunge line and train her to give to the bit like she should; but being head strong is not an easy fix. This would involve having your daughter have to step back while the horse is being re-schooled and that takes time and patience and money. But it's possible that a good trainer can get this pony up and going in time the way she should and probably worth a try, but you won't know until you get the trainer back out to do an evalution.
Here's the flip side
2. Having a horse that is head strong is very frustrating for anyone especially if they are green. The trainer should be there to try a prospective horse for the student and she would detect right off if this was a good mount to invest in. I noticed in a picture that you posted that the horse was posed with a chain on her nose and I wondered what was up with that. The child can get so frustrated then it becomes a chore to ride instead of a happy time. The child can become frightened and once the horse realizes that, many horses will capitalize that and then take it to another level and who knows what will be next, if anything. If Cassidy is already apprehensive about riding Philly, it's not going to be an easy road getting her to ride another horse if her confidence has been compromised. Before you know it, she might say no more horses and be soured from the whole thing.
You might want to consider that if you sell Philly not to buy another horse at all for Cassidy and just put the money back into lessons and let her continue to be schooled for a year or so at a good reputable stable, even if it means traveling a distance.
The bottom line is that Cassidy needs to be with a trainer with or without her own horse. Things like this happen all the time and you are not alone so try not to feel so badly about it. Best wishes in whatever you decide to do during this difficult time and big hugs to Cassidy from the mountain.
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