Overall it looks like a solid training level ride.
You would do better to post THROUGH your hands and not break at your wrist to punish him on the bars of his mouth. You DO want a nice straight line from your elbow to his mouth. This will help you with the transitions and help with that little bit of tail swishing and I suspect him pulling on you.
Your leg looks good. I'd use it more at the girth rather than moving it back to give a cue in trot and carry a whip to back up your aid until he's responding properly.
He looks like a horse I'd spend 30 minutes walking on to get him soft and yielding. Little bits of lateral work with minimal pressure. You want him to learn to wait for you. He's got his head bent, but he's not through in the back. Rather he's rushing and the hips are just trailing behind. Start to mobilize the different parts of his body. Do turns on the forehand and practice keeping the shoulders where you want. Some leg yield and SI are also probably in order. I like to spend a few minutes with a rushy or stiff horse just "rolling their crest over" but be careful- if you are not actively listening to your horse and responding appropriately, this can turn into seesawing.
In your canter transitions it's a bit of a mess, which is fine and typical with something green, but you don't want rushing to become a habit. If he's just being a butt- ask nicely for your transition, and when he pulls that running thing goose him hard with your outside leg or give him a sharp tap with your whip. You want when you ask for canter to get an immediate response. So let your correction aid be large enough to get his attention and get him immediately into the canter so you can tell him right away what a good boy he is and reward him. Wash. Rinse. Repeat.
If he's unbalanced and can't get up into canter without running because he's physically not ready, that means you need to work on his response to your half-halts, your timing (so he has the best chance to go well when you give the cue.), And how you prepare him. This horse needs a lot of suppling work and to become loose in the back. Don't worry about his head. It won't fall off while you go back and fix his body.
Overall you have a nice seat and position and he's a neat horse whose making a good effort.