I don't think they can re-train the horses because they have little money and not that many people work there. They tend to try to train them not to do things in lessons.
I don't think he does it to stretch, it is an insistant tugging and it stops as soon as I let go of the reins. I asked my instructor what I should do when he does it, and she said let go of the reins.
I'm not allowed to give soft contact, I am told to tighten up my reins a lot, more than I think I should, when I go into trot and even more when going into canter (probably because the faster he goes, the harder it is to slow him down). Once when I was trying to slow him down I relased a bit and I got told not to do that until he has done completely what I want him to do (like if i'm going from trot to walk, I should only release a bit when he has gone to walk, not if he slows down in trot a bit)
But to he honest, I haven't really been taught about release as a reward. All I am told it to tighten up my reins. I only learnt about release from this forum.
In which I also agree highly with that smrobs said.
Unfortunately, from what you've said, your instructor has been enforcing this behavior in the horse for quite some time. Whenever he pulls and you throw him the slack in the reins, that is effectively teaching him to brace on the bit and pull constantly. So, that makes him harder to collect, harder to stop, and harder to control. It would likely take an experienced trainer and quite a long time to correct this behavior. BUT, even if it was corrected by a trainer, unless your instructor changed her teaching methods, the same problem would come right back.
I think you shouldn't be in group lessons.. because you aren't getting the instruction you need. Also I think you shouldn't be riding with this instructor and her horses, because she's teaching him HORRID habits that will most definitely making riding hard and not fun and frustrating for you or the horse.
Personally.. from where you are at now.. I would much rather see you lease a horse than take these lessons. Some people that lease, offer to give lessons or some horses are huge babysitters that take care of their peeps. Or maybe you could start doing chores around the house to afford both.
They have a saying.. "perfect practice makes for near perfect." If you continue to have cruddy lessons.. you start riding cruddy.. and then every horse you touch starts being cruddy with you, instead of soft and giving and responsive and willing.
It's hard to convince non horsey people what is correct and what isn't--for the case of your mom. Mine thought I was being taught wonderfully for yeaaars. Little did she know how terrible the instruction I got, how dangerous, and how habit forming it was. Lucky for me we moved so much that I never got past trot. But even then I didn't know how to trot until I started volunteering at a therapeutic riding center.. and I learned everything! (except canter and dressage movements..)
Find another stable where you can take private or semi private (2 people max!) lessons so you can get the instruction you're paying for!
Seriously, I have a horse and I would totally teach you with him how to ride. I'm sure there are many other horse owners too with horses to lease that would do the same.
=/ sorry you're in this situation.. just try to get out of it. Maybe research and find some other riding schools and email them or call them and just talk about it. It'd help your case with your mom and by talking it through with the possible instructors, you can find one that is willing to help you become a rider.. instead of a frustrated passenger.
Hopefully I wasn't rude.. I don't intend to be! :)