Oh man, you poor bastard.
Thats a real tricky problem you've got yourself into.
Are you able to use the hand at all? to say use your wrist to push a control stick?
Auto is a must, then use your damaged hand to push on control stick forward and back for acc and brake, then steer with your good hand.
You'll have to sit the controled on your lap though, and maybe stick to driving the slower cars.
If you can't use the hand at all, then it becomes tricky....
Again Auto is a must... ummm, ok, Use R1 and R2 for accl and Brake, and then use the R analoge stick for steering.
Good luck, and I hope the hand gets better fast.