If you’re under the (typically, false) impression your horse cannot pick up that shoulder and give you a clean, light turn on the haunches – you might want to re-think the situation. It usually is not that the horse can’t physically do it (unless it's a skeletal/muscle issue), it is often that he won’t do it – either (1) due to lack of respect for his rider, (2) due to tension, or (3) because he has never been taught to be light.
1 - If it is a lack of respect that is the issue, the job is yours to earn your horse’s respect – play games on the ground that get your horse moving his feet more than yours: moving his front end around, moving his hind end around, backing up. Use assertive body language and increasing phases of “ask” until you get the response you desire. The Parelli 7 games are one great way to earn respect, roundpenning with plenty of changes in direction can be another. You can also earn respect under-saddle by working on impulsion through transitions, etc.
2 - If the issue is tension in the horse – circles circles circles – work on all sorts of patterns that include bends and circles. Keep your own body relaxed and free of tension. Earn your horse’s trust (not only in you, but in your leadership) and be consistent and fair. Challenge your horse and work on some wet saddle blankets! Use exercises that encourage relaxation and suppleness (Jane Savoie has some fabulous exercises that address such) so your horse does not 'lock up'.
3 - Teach your horse to release to pressure by asking him, first on the ground, to release to various points of pressure you apply on his body. Ask in increasing phases (hair, skin, muscle, bone) and release the instant your horse responds. Have long phases to start, and shorten the phases as your horse understands your request and thus as you ask more of him. Keep yourself light and always allow him the chance to be light!
For the horse who is leaning on your hands – quit giving him something to lean on! It’s as simple as that! A leaning horse is the result of a vicious circle - horse pulls, so your hands brace and harden, horse leans further, your hands toughen up even further...and so it continues. It's a two-way street. If you take away the contact, your horse no longer has anything to lean on. The first thing we did for my mom’s OTTB when we first got him off the track, was to remove all bit contact under-saddle and ride him on a looped rein. It was a little hilarious, because he felt like a fish out of water, floundering about as he tried to figure out what he was now supposed to do. No one was directing his every (leaning) step!! He had to take responsibility for himself, think for himself, and be light – there was no longer anything to lean on! He was a little confused at first, but quickly figured it out. Once he had a solid foundation, we then took it back up to the next level – contact. This is not to be confused with asking the horse to pick up the bit, which comes as a result of progression along the dressage training scale; picking up contact on the horse who is ready for it consists solely of picking up the slack in the reins to the point where you can subtly feel his mouth at the end of your reins - that's it. At that point you start developing the horse to where he eventually picks up the bit of his own accord, as a result of the exercises and patterns you're asking him to perform (using bend and inside leg to outside rein). For the less enthusiastic or established leaner, you can do two things when he goes to lean: close your legs and close your hands. Your legs should gently but abruptly bump him, effectively driving his hind under him a little more and causing him to pick up his front end. Simultaneously, close your hands abruptly as he reaches down to lean, allowing him to "hit the end of the rein". Lateral work and poles on the ground - exercises that require increased engagement, will also prevent your horse from leaning.
Please make sure to not confuse a horse reaching down and stretching over his back with a horse who is leaning - a horse who is leaning is putting a lot of weight in your hands, you will feel your arms and shoulders being dragged downward. The horse who is stretching over his back should be doing so while remaining engaged (at least mostly) and should be light in your hand. If you're not sure what your horse is doing or why, consult a professional.
Many times I feel horses are thrown into contact work without sufficient basics. A horse should be light on a loose rein first, should be responsible for himself (ie, maintain gait, self-carriage to a limited extent, and balance), and should know how to think for himself, first. He should be very responsive to weight shifts, to leg aids, and should be soft, supple, and relaxed. There should be no resistance on a loose rein and there should also be no dependance on that rein for control; it should be an intimate means of communication! Only then should the slack be taken up in the reins and the horse progressed up the training scale. At that point, the rider is solely taking up slack in the reins - not pulling on the horse's mouth or trying to initiate contact by the horse or what.
Another tip for the leaner is: impulsion. Transitions as well as changes in pace within a gait (such as a trot) work to get the horse working more off its hind end. A horse on his hind end is, obviously, not on the forehand and in a position to lean. A little squeeze when your horse goes down to lean forces him to step under himself better for a moment or two, which causes the poll to come up. In dressage, you work to the point where eventually the poll is the highest point in the collected horse - obviously he is not leaning on your hands by that point! You don't have to be a dressage rider to achieve the same lightness and level of collection, but you can do the same or similar exercises. Really effective can be to squeeze and bump gently with your hands simultaneously, as previously mentioned. Don't nag at your horse, but only correct him when he actually makes a mistake. On the note of exercises, I have found the book: Progressive School Exercises for Dressage and Jumping, by Islay Auty, to be extremely beneficial - I use it on a constant basis, whether working my western horses or my dressage and jumping prospects. A specific exercise you can do is to set up 4 pairs of cones, one pair at each "corner" of a 20m circle (creating a "+" or "x"). Space each pair of cones 3-4 feet apart, along the circumference of the 20m circle, with the goal of riding between each pair. Start by riding the circle at a walk, then progress to a trot, then finally a canter - don't progress to the next gait until the former is well-established. Ensure your horse is bent, supple, relaxed, and balanced going around the circle. As you ride between the pair of cones, slow the horse (if you're working on contact, don't pull back, just close your hands and relax your seat). As you leave the pair of cones, open your hands and allow the horse to move out again (you should just have to open your hands, if not, give a light squeeze - if impulsion is a problem, play some Point-2-Point or other such simple exercise). The goal is to teach the horse to sit back on his hindquarters and work off his hind end - by slowing he sits back in the first place, by asking for impulsion afterwards he works off that hind end. With the western horse, your hand and seat cues are essentially the same, even if you are working off a loose rein (if your horse has trouble "hearing" you, use heavier reins or shorten up the rein length).
Keep in mind also that the horse who is leaning is likely doing so due to lack of strength and balance. He is using his rider as a 'fifth leg' so as to balance, as opposed to using his hind end. The ultimate object is to work the horse in such a way where he has to increasingly engage (ie, lateral work, poles, hills) so he develops strength and the habit of carrying himself in a more uphill manner. Horses with a downhill build (ie, top of butt higher than top of wither) or with a low-set neck (ie, below the point of shoulder) will have a stronger tendency to lean and will have more difficulty rocking back onto their haunches - be patient! Such a horse will have limits to his ability to move in a more uphill manner. Horses, just as people, can also be lazy, which means they are not driving from behind and engaging as they should be. Same principles as previously mentioned all apply.
Last but not least, our horses are a reflection of us, so a tense body or hard hands = a hard mouth and/or a heavy horse. Keep your body and hands soft!! There is many a time I have had to consciously remind myself to lighten up my hands or legs – instantly the horse I am riding would lighten up as well. Tension in your own body can also reflect as tension in the horse too. Often, too, we just have to give our horses the chance to be light with leg aids - they sometimes surprise us! Always be as light as possible but as effective as necessary - make your corrections to be effective but always return to being light. On a related note, be sure to maintain correct position, as a leaning rider will also create a leaning horse who is on his forehand.