(sqeeze, click, spank- it is a Clinton Anderson thing but not sure how to appropriately attribute it to him)
Sqeeze with your legs (whatever elg cue you want for canter)
If the horse does not respond, do not sqeeze tighter, but click with your mouth
If the horse does not respond, spank it, lightly in a rythm of one-two-three-four and with each set of four get more intense (use the ends of your rein or your crop)
Release all cues IMMEDIATELY for the little try of a canter. Over exaggerate- take your legs way off the horse, shut your lips, and keep your hands light. Your horse will learn quicker by how fast you release. Do this until your horse understands that the leg cue means canter.
Right now do not worry about leads. Your horse will figure that out and first things first.
Now to get your horse comfortable, cue and elevate if needed for canter. Be in an arena if possible. Let your horse decided where to go- DO NOT STEER, only keep it going at a canter. Let it completely drop stride before you cue and elevate cues..... sit on your horse and just RIDE. Concentrate on looking where your horse is going and getting your seat deep in the saddle. The only thing you need to do is keep your horse going at a canter. IT is very hard not to steer, but horses can only learn one thing at a time. So for now, just canter.... do this for about ten minutes. Your horse will settle into a ratable stride and once it does that, quit for the day.
I have done this with my horse multiple times and now he not only canters off lightly without lurching, but he also has a speed he picks himself, which is slow and steady.
It is hard not to steer when going towards mud, but I just stayed on and when he dropped stride, I let him know I wanted him to keep cantering, even in the mud or next to the exit gate or next to his friends. Don't stop after a minute, literally do ten or more of cantering. Make it obvious to your horse that this is what you want.
You can worry about leads, steering ,headset, all that other stuff later. Right now, just teach canter.