My RI uses a brilliant technique to get young horses or those unbalanced in canter going. She believes that often we ask a horse to canter when we want it to, which is fine when the horse is balanced and capable of cantering on both reins as required. However this is a big ask for a young horse. So, what you do is basically ignore your normal canter aids. Later when your horse is going nicely you can reintroduce them but for now your horse needs to be able canter on their terms.
So, get your horse into a nice forward going rising trot on large (20m+ circle). Then you shout out loud:
" 'HORSES NAME' when you are ready you may canter. You don't have to if you want to but you will trot faster"
If your horse does not canter you must push them on to trot faster.
Then repeat " 'HORSES NAME' when you are ready you may canter. You don't have to if you want to but you will trot faster"
Again if they do not canter push them on faster in trot.
(I know that talking out loud to yourself seems stupid at first! But RI says if you say it, you believe it and it happens. It does work I promise)
The first few times you do this it may take a while for your horse to break into canter. For me it took over 20 mins the first time! Try to be as relaxed as possible so you don't inhibit him in any way.
Once your horse is canter, hopefully you're not too exhausted by this point!, sit as quietly and as balanced as you can and only help him to steer round the circle. If your horse breaks back into trot let him. Don't force him to keep cantering. He has broken into trot because he feels he cannot carry on cantering so let him do it. Once he is trotting again give him a few minutes quietly trotting and then repeat the exercise. Try and get 1 or 2 more canters during the session. If he is particularly poor on one rein start with the good rein to give him confidence.
If you keep repeating this exercise on both reins you should find that over a period of a few weeks it gets easier and easier. Your horse should start to canter earlier as you recite your mantra, often only as quickly as the first part saying his name. Once you get to this stage you can start to ask for canter just by saying "canter" in a nice soft voice.
This method works because you give your horse the chance to canter only when he is balanced and able to, giving him time to sort out his legs for himself. However by constantly increasing the speed of the trot it ensures he cannot escape and avoid the canter. What you will also find is that he will very rarely break onto the wrong leg. If you watch horses in a field they never canter on the wrong leg, this is because they have chosen when they are going to canter and wouldn't do it if it meant being on the wrong leg. Once your horse is strong and balanced he will be able to move his legs and pick up the canter when you ask on the correct leg but until he is ready this technique means he always gets a positive experience and result.
Once you have this cracked you can go on to using the normal aids. To be honest all I now need is a very very light aid while saying "canter" and that gets me a lovely canter in the correct leg every time.
Good lick and let us know how you get on