He looks like a lovely horse! What a kind expression and I think he'll be quite lovely with some more groceries and muscle. There are lots of different schools of thought on how to best "re-feed" a skinny horse, but do some research on your own in addition to what your vet, trainer, etc. tell you and make sure that he is getting adequate protein and fat. For the hard keeping types this can make all the difference in the world.
I also think you've got way too much gear on his head. If you are not trying to jump or do intensive flat work, can you ride him in a simple snaffle (or perhaps just snaffle +martingale)? If you can ride him just fine doing simple exercises in the snaffle, but he is heavy or "difficult" when doing lots of trot work, lateral work, cantering or jumping, it is likely because he is simply unable to carry himself properly due to lack of muscling and the previous improper training you have mentioned. Please, please ditch the draw reins. They can only have the opposite effect from what you say that you want.
If this horse were mine, I would take a break from jumping him (but definitely would incorporate ground poles/cavelleti as he was ready for them) and would focus primarily on hacking out, with hill work if possible. I would ride him in the gentlest bit and with the lightest contact possible. If he felt unbalanced or like he was leaning/bracing, I would use downward transitions, halts and rein backs to rebalance him rather than using heavy rein aids. It is hard work, but if you are gentle, firm and consistent, you can intact retrain a horse to develop the correct musculature and learn to travel comfortably in the contact. Keep the sessions short at first and increase them in length as he becomes fitter.
I take the time to write all of this because he looks to be a kind horse, you look to be a kind rider and I hope you will consider what I and many other posters have written because it looks like you two could be a very nice pair if you were willing to take a bit of a different approach.