Kayty said it very well.
My barrel mare right now is very heavy on the forehand. Obviously, in a high speed sport where the horse has to turn on their hind...That's not only undesireable but dangerous too. The first time I ran her she almost fell ontop of both of us around the first barrel.
Ontop of the octagons, squares, etc Kayty said, I like to do a lot of holding on straight lines and drive drive drive with your legs. At the trot this is my favorite, because it's easier to feel (for me at least) at the trot when they get a real extension in the stride and tuck the butt.
You ride english so this may not be ideal for you, but my other exercise is rollbacks. The barrel racer kind, the run down the fence at Mach 5 and then abruptly turn into the fence.
It's super fun but it can easily make your horse hot. You could probably do it at the trot or slow canter though. They really have no choice but to go onto their hind end or they smack their face on the rail...Just be sure to give enough room. Some people I know get WAY too close to the fence and there is no space for the horse to actually do the rollback, so they DO end up smacking their face....