If you mess up, don't dwell on that single mistake, chances are you still have a good portion of your round left. So instead, put the mistake behind you and start planning for the next part of your course you are going to execute. 😊
Hand eye coordination is basically what it is.
Off your horse you can practice this by throwing a tennis ball around - throwing it against the wall or laying down and throwing it up in the air. Juggling also.
On your horse you can do any Mounted Games games and continuing to correct your horse as you are best you can (be conscious of it and you will improve)
When you improve your hand eye coordination, you'll subsequently improve your reaction time.
Hand eye coordination is great to be good at as this also helps with being able to pick the correct striding and be able to have an independent seat.
And also, dependant on what your horse is doing for you to have a sharp response, you can learn to feel it coming before it happens.
(Dramatic example) Say he tenses his back and shortens his stride before he bucks. There's a feeling before the following action.
Just try to correct mistakes as soon as it happens and keep riding and be aware of this and then everytually it'll become natural and you won't even have to think about it.