Some of the mounted games I used to do when I was in 4-H:
Rescue race - you are at one end of the arena, your partner is at the other standing on a barrel. You race to the end, pick up your partner and race back.
Cowhide race - you are on the horse your partner is laying on an old cowhide that you have to pull to the end of the arena and back. Lots of fun ensues when you make the turn at a run at the end of the arena your friend goes flying off into the dirt - er... I mean the other teams does that! The variation is that you start on the horse, race to the other end, get off, switch to the cowhide and your partner pulls you back.
Dollar race - you ride bareback with a dollar bill under your thigh. You have to ride all the gaits and if you lose the dollar bill, you're out. Last one riding wins all the money.
Egg race - you have an egg on a spoon and you have to walk, trot, gallop without dropping the egg. Lots of fun when the egg breaks on you or horse. NOT!
Flag race - you race to the end of the arena where a flat is in the ground and race back with it.
I've got some others but that will give you some cool ideas.