Lesson horses have a tough life. They sometimes bully people because they in turn are bullied BY people. Not intentionally, of course. But because a riding lesson is all about the RIDER learning while the horse's feelings or opinions are irrelevant. "Ask, Tell, DEMAND!" is frequently taught, and if you aren't willing to DEMAND then "The horse has your number" or "He's bullying you!"
One of my three is an ex-lesson horse. 5 years later, if you try to ride him in an arena, he'll buck, run and fight. Take him out on a trail and he is a very sane, cheerful horse. Maybe the safest horse we have.
Even so, when I took lessons (for 3-4 months), there was only one horse who needed to be ridden with a whip, and you only needed it once a ride to convince him.
Here is how it looks to me. You want to be able to use your heels every stride. You are "behind the horse" and using grip with the knee instead of letting your weight flow past your knees and into the stirrups. I suspect part of the reason you need to kick every stride is that you are putting too much pressure on the reins, effectively riding the brakes while pushing on the gas pedal. Your hands are not moving with the horse's mouth. You look like you are working hard at the canter - a gait that needs you to flow
so your horse can flow under you.
I'm mostly a western rider, but I'd recommend giving the horse some slack - or going bitless. Too much hand, particularly since the hands are out of synch with the horse. Stop riding without stirrups and pry your knees apart, if need be, so your weight can flow past your knees and into your stirrups. OR
learn to use more seat and less knee. The horse isn't bullying you. He's frustrated.
Yes, I'd lower the stirrups.
You might also consider Common Sense Horsemanship
Disclosure: I'm a backyard rider of backyard horses. We ride on trails and rarely in the arena. I don't teach (no one would ask me to
) and I don't show or compete. I'd never win an equitation contest. Probably be thrown out. From yesterday's ride - a very different world than yours. If I've been offensive (or grossly irrelevant), I apologize now.