This is most likely going to be solved through lots of flatwork. I’m going to give you an idea of where to start, and then you’ll need to develop a flat program with your trainer.
Flatwork is NOT what most hunter/jumper/eq (and heck, even some event) riders do: pull horsey out, w/t/c around the ring on the rail, reverse, done. Repeat until jump day.
Flatwork IS working your horse in a manner to develop strength, suppleness, adjustability, and something I’ve heard best described as “Broke-ness” (this concept is one I heard from an old cowboy about cutters and reiners, but definitely applies to the English disciplines: it’s getting the horse to the point where with your feet, seat, and hands, you can move each part of the horse independently: put their feet where you want them, keep the head where you want it, lift the back and/or the wither without changing much else, adjust where the poll sits, etc. all quietly without a battle between you and the horse).
This does NOT get done with laps around the arena, and resist the temptation to jump straight to “fun” stuff like advanced lateral work, half pass, and pirhouette. Some of the most difficult lessons I’ve ever had happened with dressage trainers (and my jumper trainer, with an extensive dressage background) at the trot on 30m circles- and you guessed it, they focused on the idea of “broke-ness” explained above. Learning pirouette-once I had a firm grip on the basics-was WAY easier.
What this will do for you is give you a horse that is strong enough to hold together through jump courses and combinations in addition to a horse who is trained enough to really listen while on course- and you will know what to ask your (listening) horse what to do. Instead of half-halt no response, you’ll get half-halt horse lifts front end, lowers hind end, engages back, shortens stride just a little, pats the ground and lightly jumps over the jump.
It is seriously all in the flatwork. If you already have an athletic horse who likes jumping and is careful, you will not solve this by jumping him into the ground.