What difficulty are you playing on? On Normal it should be pretty easy to beat Honest Tom, just keep running (default key Shift, if you're using the keyboard); if you're getting behind just release Shift and tap it again, it should give your horse a boost -- just make sure not to work your horse to death; if you keep tapping Shift your horse's health will keep going down until it dies.
I'm saying this because I've come across several people who didn't understand how running works, in this game; most of them think it's just a temporary burst when, in fact, you can keep the run key pressed and the horse will keep running: you don't even have to press forward, just steer left and right. If you want to get boosts while running, just release the run key for a second and press it again -- you can do this as many times as you'd like, as long as your horse has the stamina/health for it.
There's also a shortcut in the race, if you're still having trouble beating Tom; it's right before the turn that leads into the log jumping: just head right along the rocks instead of keeping on the track, and reenter the track after the turn. It's not that much of a shortcut, but it's almost mandatory if you're playing on Hard.