Here's a question I got today about a horse who leans on the bit. My answer is below: I rode a horse this afternoon for a lady who indicated he had a tendency to bolt off when cantering. He is a 17 year old warmblood. Previously was trained as part of a four-in-hand combined driving team.
I didn't canter him today since he was not in balance at the trot, but it appears that he more or less just has a tendency to get stronger and stronger at the canter and pick up speed. Not really a reactive bolter or anything like that. His owner cantered him a little bit before I hopped on and he didn't do anything out of the way.
He is VERY heavy in the bridle. If you drop him he doesn't really speed up he just gets strung out. He's a smart boy and seems willing enough. Let me know if you have any recommendations on getting him lighter, I can't recall working with one who leans this bad. He's built pretty uphill and is a nice mover, more balanced than I expected. Needs some fitness and some more strength in the hind end, but I don't think that will completely resolve the issue. I couldn't figure out an effective half halt today. He is not super hot. Horses were running around the pasture next to him and he wasn't bothered. He moves off the leg well and is easy enough to maneuver the various parts. Brakes don't work so well. Can't seem to get more than a step or two of self carriage at a time. He's more fun than a gym membership.
An effective half-halt
I think you put your finger on the problem when you said you could not figure out an effective half-halt. He may not have one. I would start there and teach him one.
After a little warm up, do some walk-halt transitions. When he halts, give the rein a half-inch forward so he is standing in balance with no pressure. Then reconnect with your heavy, soft, elastic elbow as you walk on. When you ask for the down transition make sure you have a soft, steady connection in your hand and that your seat is following his motion. Then raise your collarbone briefly in a little pulse that says "wait." Don't stick in it, just use the pulse of energy to let him feel your seat balancing him more uphill onto his haunches. Do one or two pulses and then close your fingers and stop your seat as you do the final pulse and halt. Try not to back your hand. Instead ride him into a fixed hand. Then give the half-inch release and let him stand for three or four seconds. Praise and repeat.
If he starts to get crooked as you ask for the halt, walk on and straighten him, then try again. He doesn't have to be super straight in the halt yet. Just try to get across the idea of listening to your seat. If he ignores your pulsing collarbone lifts, try closing your thighs a little more against the saddle so that your body mechanics transmit to him during the half-halt. Don't get stuck in gripping with your thighs though. Release and be light again once you make the pulse to "wait." His response will come through more after the release. You also want to make sure he steps off from the halt with energy. Don't let him loaf.
Next, do the exercise using trot-walk transitions. In the rising trot, make the half-halt pulse at the top of your posting stride. Close your thighs/knees a bit and lift your collarbone, thinking "wait." Your hands should continue their usual job of softly connecting and suppling the horse. Only close the fingers and fix the hand briefly when you feel the walk is ready to happen. At that moment, sit and swing the horse into walk, releasing the fixed hand and following the horse's motion. If he runs through your hand, bring him to a full halt and release, let him stand in balance, then try again.
I like to do these transitions on a 20-meter circle too. There are several variations that help teach the half-halt. One I like in particular is, in the rising trot, to play the short-trot/long trot game. Go a bit more forward for half the circle, then use your half-halt "pulse" a few times and come to a shorter trot. Remember to use your body first, not the reins. Once you get a balanced shorter trot (not slower, but shorter steps), go forward again. The hands should always be doing the same job of suppling (with a rotating inside rein) and connecting (with a steady, elastic outside rein) the horse during the half-halts. When you are ready to make the actual down transition, the fingers close and the hand is briefly fixed while your seat and leg support the horse to step his hind legs under his body mass while slowing the forward motion. Remember to change directions and rest often with a big free walk. This is hard work for the horse.
For the half-halt to be successful, the horse needs to bring energy forward from the hind legs over his back and connect to the rider's hand. Then he has to be attentive to the rider's request to make the front end "wait" while he brings his hind legs more under his mass. If you only ask for him to "wait" using the reins, he has several avenues of response. He may just slow down, or hyper flex and drop the connection. Or, lean on the bit, as you have indicated this horse does. Because a lot of his training has been as a carriage horse, I suspect he is not used to being attentive to the rider's seat.
What I like about teaching the half-halt using the collarbone pulse is that it allows you to use one of your most effective rider tools: the long lever your body forms from your seat to the top of your head. We learn to be elastic and stable in our torsos, spines, and shoulders; to carry our heads erect; to let our arms drape off of our shoulders. Then we have this incredibly powerful lever we can use to help balance our partner. Mostly, we focus on keeping the various parts of that lever independent and elastic, so that we don't block our horses from moving. But in the half-halt we want to use our lever more in one piece to ask the front end of the horse to wait while the hind end catches up. This dynamic, when we are successful, creates balance and lightness in front. The reason I like to use the pulse, or stretch, of the collarbone at the top of the posting stride is that that is the moment when your lever is most fully extended and can be most influential. So, this "pulse" involves a bit more stretch of the rider's torso on the front of the body, a brief lifting of the collarbone in rhythm with the horse's trot, a closing of the rider's upper legs to transmit her message to the horse, and, finally, a release. In general, I think of my upper body as influencing the front half of the horse, and my seat and legs as influencing the hind end of the horse. A successful half-halt comes from the correct use and coordination of all these aids, as you very insightfully indicated by your statement that you were not able to "figure out an effective half-halt today."
I have ridden a number of upper-level horses who did not have a clear and reliable half-halt, and much of the training I focus on in a horse's education is to create a good half-halt. I think you will be very helpful to this horse as you proceed to teach him one.
Let me know if anything here is unclear, or if you need additional exercises, etc. Good luck and happy riding.