Definitely have a good think about the walk-canter transition. Think it out in advance, thing about what 'shape' you will ask Pete to be in before you apply the canter aid (Bent round your inside leg - haunches in a little). Now think about when you will apply the aid. For joePony, I need to be shaped up, in position, then as his Outside Fore reaches forward, I apply my outside leg strongly to push His outside hind under him and strike off correctly. You might be quicker than me (I'm old, you're young, don't be rubbing it in) ... and you might have to wait til the horse is obviously moving forward over his Outside Fore before you apply the aid.
The other thing to think about is the request-response training cycle here. How can you make it a Real Good Thing for Pete whenever he gets it right ? Because this is a thing that some person has messed him up on, it is not the sort of thing that you will easily correct by penalising the wrong response*. You're going to be much better off here rewarding the right response. So, when he gets it wrong, each time ... what's the 'reward' ? More work. Back down to a walk, and without a break, shape him up to be a banana, wait for the timing, and into a canter. Now - just one or two steps - he's either correct or not - either way, slow and stop ... and if he gets it right - you have to be super-alert to that and praise him out of all proportion. Get your Mom to run over with a carrot. Hug him til he's embarrassed. If it was me, and joePony, I'd be trying this after he'd been well warmed up, and the first time he got it right, I'd be in a flying dismount, ass in the air even before we were down to a walk and put him straight up with a big feed and give him the rest of the day off. That's how strongly I feel about rewarding the try, when the horse is trying his best and still getting the wrong response. He needs to really know how much you appreciate his thinking it out.
Then in future sessions, ok, I'd ask for a couple of correct responses before putting him up. But only advance it very slowly. In this way you develop a horse who has a strong incentive to figure out what you want - and to do it !
* unlike, say biting, where I find a bit of negative response to his wrong behaviour to be quite acceptable.