MIeventer: I didn't mean that he was doing anything wrong, I meant so that he gets both his feet picked up. In my opinion that's how to teach them to jump is lunging them and free jumping them. Is this not correct??
A good way to teach a horse to tuck up and round over fences is by doing bigger fences, in grids.
The OP has only been jumping just very minimally and is just starting out. She is more focused on her form, which is very important especially at the early stages - and I wouldn't be focsed on the impulsion or asking for a more forward rhythm from the horse at all.
When the rider is ready, at that point - she will beable to close her lower leg around her horses girth, and use her seat for a more forward momentum out of her pony.
But right now, I wouldn't even be worried about that, nor what her pony looks like going over a small x rail - if I were her coach, I would be totally 100% focused on what she is doing in the saddle form wise.
Solidifying form now, at an early stage is a fabulous way to give the rider the essential tools needed for when she is ready to merge forward in her training.
The Lunge Line is a great idea - personally I would put the OP on a lunge line with her pony - so that the OP can focus directly on herself and where she is in the saddle, instead of having to not only worry about where she is, and where her pony is at the same time.
I think she's doing a great job as it is for her 2nd time only over fences.