Many times when a horse has trouble taking a lead, it is due to the fact that it is stiff and needs to have help becoming supple through the shoulders, ribs, and hind end. You might want to find a professional who could help you through this. Sometimes we also hinder the horse by leaning over the inside leg, which does not allow the horse to pick up the inside lead. You can try working the horse on circles and working on developing a soft mouth by asking the horse to bend through the neck. Ask on a circle for a lope/canter with your outside leg moved farther back than normal; use a kiss or some kind of verbal cue at the same time. If the horse picks up the incorrect lead, keep it on the circle, then ask it to come down to the trot and ask again. Continue doing this, and the horse usually will pick up the correct lead with several tries. Once the horse picks up the lead, then lope it for several minutes on that lead; then stop it and praise it. Again, if the horse is too stiff, you may need someone else to help you through this.

Additional information:
Rider's Body Position
To get a horse to take the correct lead at a canter or lope, the rider must first realize that they need to sit back and avoid leaning over the horse's shoulder to see the lead by looking at the horse's front legs. Sit back in the saddle, and place your weight on the hip on the opposite side of the desired lead (i.e., if you want the left lead, put your weight on the right hip and vice versa).

Rider's Hand Position
When riding either one- or two-handed, be sure the horse's nose is tilted slightly to the outside of the circle and away from the lead you wish to take (i.e., if you want the right lead, then the horse's head should be tilted toward the left). Doing this will free up the leading leg and shoulder to step forward first, and you cue your horse to canter and will thus increase your chances of the correct lead.

Rider's Leg Position
When applying leg pressure to step your horse off into a canter, be sure to apply the most pressure with your outside leg or the opposite leg of the lead you wish to achieve (i.e., if you want the right lead, then apply more pressure with your left leg). Make sure the pressure is consistent and solid until the horse steps off into the canter. If the horse leans on your leg or refuses to canter, apply more pressure or rhythmic kicks until the horse moves off your leg in the opposite direction.

Horse's Body Position
As you keep your body in position while using your legs and hands correctly, your horse will keep a straighter body alignment too. it is important to keep your horse's shoulder from dropping while cantering, and the application of outside rein and outside leg will help hold your horse in the correct lead.

Exercise
If you are still having trouble getting your horse to take the correct lead, send your horse off into a canter, and then as you are cantering, turn your horse around and start cantering in the other direction whereby the lead the horse is taking is now correct. Some horses just need to be taught what is correct in terms of leads, and this would be one way to get your horse used to cantering on the correct lead when it would normally not do so.

The horse may be stiffer in one direction than the other, and you may need to implement flexing and bending exercises to teach your horse how to move its shoulder and hips on your cue. This is best started at a walk and then progressing into the trot. If you can control your horse's body at the walk and trot, then you will be more likely to do so at a canter/lope and be able to achieve the correct lead every time.