Do you have an instructor/trainer and do other people ride this horse? Repeated spooking, turning and trotting off is the horse telling you they don’t care to work any more and leaving and you not using your aids ( rein, leg, seat) to a) keep him focused on you to block the spook and b) you not using the aids to correct and clearly direct him if his attention does wander and failing to correct his actions so he learns he can’t just leave if he feels like it.
Hes learned he can do this because his riders have rewarded his disobedience and let him decide when and where he wants to work. Or not work. Because he’s not really bolting off and/or spinning you right out of the saddle, don’t think he’s trying to hurt you, he’s just changing the subject with you so he can keep control. And it works very well for him to call the shots on when and where he works. His riders reward him by letting him, every time he gets away with it, he wins.
Dominance is not really the right word, that’s more for ground work. You need to communicate to him via the aids and back it up with correction when he blows you off.
The time for action is BEFORE he stops. Leg to go forward, pick him up with the hand to block the direction change and seat firmly in the saddle to give you the strength to keep the leg and hand where they belong. When you feel like he’s going to stop anyway? Stick behind the leg… and don’t tap, tap, tap, smack him and leave him alone. You want to use the stick to reinforce your leg aid and telling him to go forward, stick is not an aid to direct but a correction to back up ignored aids. Hope that makes sense.
You telling us you tap him with the stick after he has already “ spooked” and is trotting off towards the barn I’m guessing tells me you need to take a few flat only lessons, specifically in using the aids to keep the horse forward, focused on you and to block a spook in the first place with keg and rein aids and let him know he’s made a mistake and correct him if he blows you off.
Its an old school horse trick also seen in novice “trained” horses. Usually the resident Attitude Adjuster fir school horses when I boarded in teaching barns, hardest thing was getting them to pull the stunts with me. Second I picked up the reins and put a leg on them, they knew better.You need to refine your riding with some flat lessons so he knows better with you. Get with your trainer/ instructor,
Keep in mind if he’s got other riders, you probably will have to remind him you won’t put up with it every time you get on.