I certainly would not be jumping after a year off. I would be doing alot of longeing to condition, and with that in mind, if the OP has the horse rhythmic on the longe, she hasn’t reported how he is approaching and handling a jump on the longe. If she hasn’t at least looked at him on the longe, I’m not sure how she can describe this as a problem. IOW, if he is not rushing on the longe, why is he rushing with his rider.
Also, I agree with the people talking about letting the horse take himself over the jump. Sure, its risky, the horse may well plow into the wall after, or plow into the ground. If he does, you can be sure he has become habitual about leaning on that bit and on the rider’s hands. He can’t lean on the rider’s hands if she doesn’t give him something to lean on /pull on. I guess for me if a horse takes hold like that I habitually let go. They usually stumble and sometimes go down. They are not supposed to be leaning on the bit, and its a sign they are completely on the forehand and not using their hind end, so there is some moment that the rider is not addressing when the horse moves to the forehand before any of this happens.
I think this is the beginning of any kind of retraining on how the horse uses contact. But certainly, if he blowing through the bit, he can only do it if there is resistance there against which to blow. If that makes sense.