I ride a big-part draft who used to have this problem as well. She would cut halfway across the arena, drift inwards... and this was just on the flat. Add jumping to the mix and it was a complete disaster. I used this exercise to train her to go into her corners and now she's near perfect with them (keep in mind it does take time and patience, but will pay off):
Walk him into the first corner. Half-halt when you're a few strides away and then stop him when his nose is almost touching the wall. Then turn and do the same in the next corner, and so on and so forth. Pick up a trot and repeat the exercise (ride to the corner, half halt a few strides away, and stop him when he's right in front of the wall). Then do it at a canter. Don't worry about the way he'll bending, just focus on getting him into those corners. Meaning if he drifts in, pull him to the corner, use tons of inside leg, whatever you need to get him there.
Eventually he'll learn to anticipate that a half halt when approaching a corner means he needs to go into it and wait for your cue to turn. Once he learns this, half halt as you approach the corner, but instead of stopping him apply leg pressure and a touch of inside rein to bend him around your leg and keep him going. Same goes for jumping. After you go over a jump, ride to the wall and stop him there until he learns to go into the corner.
There is another easier trick you can use, but it's short-term and not so pretty looking. When you get close to a corner, lift both reins a good few inches higher than normal off the neck and move both of your hands hands towards the outside. This creates a barrier with the inside rein while keeping your horse's nose pointing straight with the outside rein. Also move your inside leg back a few inches and apply pressure. Works like a charm to get horses right into their corners but again, not so pretty looking.
Hope that helps. :)
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