Obviously there are other riders here with more experience than me, but I learned a little something about this as well. I usually ride Western, but I was taught that this could work for English as well in getting the horse to pick up the correct lead and then letting him have his head as a reward so that he knows he did the correct thing. He enjoyed cantering, so by letting him go, it was his reward.
In Western, I was taught that to help the horse pick up the correct lead, I would bend their nose to the rail and apply pressure with my outside leg as a driving force, then apply both legs when the horse started to understand and I wanted to get him up into the canter. Tipping the nose, or the head as a whole, controls the front end of the horse. By tipping their nose to the outside, the horse is inclined to pick up the inside lead.
Imagine that you are riding your horse on the left diagonal and you want him/her to pick up the right lead. By tipping the horse’s nose to the outside, you are making it more work for him/her to pick up the left lead because he’ll be throwing his body forward in a way that makes it more work. The horse will eventually figure out, with time, trial, error, and patience, that when their nose goes to the outside, by picking up the inside lead, it’s not as much work.
With the young mare I ride now in my program, she had a weird time picking up the inside lead when my instructor asked me to tip her nose to the inside for a bend. Instead, at least for the first few times, I tipped her nose to the outside to help her figure out that throwing her right shoulder in was less work. I still do it at times when she needs to figure it out, but she has gotten to the point where now she knows which lead to pick up, even when I asked her to tip her nose toward the inside.
Things like this just take time and patience. Like some other riders suggested, it could be a different behavior problem, but I’ve dealt with horses as well that were sound and just didn’t want to pick up the correct lead on occasion. Perhaps, in addition to tipping the nose, you could also spend more time stretching and working out his ‘off’ side of the body, the lead he doesn’t want to pick up. It’s possible that one side is more in shape than the other and it’s just more effort for him that he doesn’t want to deal with.