What, missing an easy shot? I expected some Grey's Scouts war footage, how disappointing...if he was standing in the stirrups at a gallop, and killing a speeding lion with his first shot-that would be nails. We do that kind of thing all the time round here.
The first thing is to do an awful lot of training with the horse, preferably from the ground before ever trying it from the saddle.
They have an excellent fright and flight reflex which means that if anything happens out of the ordinary they don't think, they run, bleedin fast for at least 450 metres before they will slow to have a look at what it might have been. A gunshot would cause that sort of reaction.
In another lifetime I have had some experience with training a horse to accept a stockwhip being cracked from the saddle. That gave some "interesting" reactions before they became accustomed to the sound.
Long, long ago in Africa, I used to borrow a couple of horses that were trained up for military patrol use. You could fire a FAL from the saddle, or even with the horse made to lie down as cover - just as they trained them to do back in old Empire cavalry days. There was even a word of command that would cause the horse to "freeze" so that you could take a better aimed shot. They didn't even seem to flinch at the noise; I guess they can see the rifle in aim and thus anticipate the shot.
(Unfortunately, I later pissed off one of these horses big time by nicking its ear with a sword whilst trying to sabre a line of paw-paws stuck onto sticks. After that incident, it went out of its way to deliberately fnuck up everything that we tried to get it to do...)
If you are going to try and shoot from horseback, do practise it quite a lot. There is nothing more embarrassing than trotting out one day and meeting some like-minded individuals and hearing "What happened to your horse's ears mate?"
Or so a friend of mine tells me. Hello Nick? how's the fly-fisihing coming?
Tom Mix, Roy Rogers, Gene Autry, Ben Johnson and John Wayne, not to mention Audie Murphy all used to manage it quite well. And hit everything they aimed at.
Coy Commander 'A' Coy my old regiment came up with the absolutely brilliant idea of equipping camels ( and riders) with gpmgs in Libya. Since we had AFV432 there I have no idea why, but he was the boss. It was very interesting, especially as a non-rider/gunner. Most amusing for ages.