Strengthening my one-sided horse...

My horse is a bit one-sided. He canters fine on his left lead, but his right lead is a little weak. He can't get into an even rhythm, and often breaks from the canter unless I keep kicking him. He's the same whether I ride english or western.
I don't notice his off side as much while trotting. He can be a bit choppy at first, but after being fully warmed up, I don't feel a difference.

He's checked daily for tenderness and any health issues, so that isn't the cause. I'm thinking it's probably because the cold and snowy weather keeps me from exercising him as much.

What are some exercises I can do that will help strengthen that side? I've been doing circles and serpentines, but I'm not seeing a difference.

The best thing to do is lots of trotting. Trotting is easier for the horse then canter work is when he has a weak side. Do lots of up and down, from walk to trot, from trot to canter. Up and down. You will have to do this every day even in the round pen if you can't ride. If he does not work the one side it wont get strong. It can take a few months to get the side back up to how it should be. Give him one day off during the week.

Spiral in, and spiral out on right lead. Lots of trotting on both diagonals for long distances with changes of speed.

If you know a decent groom, they should be able to check for muscled tightness in the area of the right shoulder/forearm.

It is very hard to keep a horse in good condition in the cold and snowy weather believe me I know how you are feeling. Even just walking on a lunge when the ground is too slippery for anything else will help him balance and bend in that direction. Work on getting him to walk in smaller then bigger then smaller circles on the lunge making sure he is really bending into the circle and not pulling his head or bulging to the outside. I always work the weak side more than the strong side (about twice as much). When the weather is good start working on transitions. Lots of up and down transitions from walk all the way up to canter and back and forth in between will really help him build muscle. Make sure when you are circling that he is bending into the circle and not just keeping his body straight while turning. Again while riding work twice as much on the weak side than the strong side.