Well it's better than you first video, but I honestly would not
be loping the pattern with him yet. You are allowing him to do some very bad habits that are only going to hurt you in the long run, because your ingraining those bad habits deeper. I would strongly recommend you to only trot at this point. No, it's not fun to go slow, but if you want a barrel horse that runs the pattern correctly, you've got to.
Going to the first barrel, at least you got him to pick up the correct lead. That's good. I am concerned though about how high his head is in the air because he seems to be evading the bit.
I don't know if you noticed, but right before the barrel turn you kinda jerked back on him a little with the reins. It's good you sat deep in your saddle, but make sure it's your body doing the rating and not your reins.
You still let him go straight into the turn without any bend in his body. Not good. He doesn't have any bend and he doesn't engage his hindquarters fully. Lift with your inside rein (twisting your wrist up and toward your opposite shoulder) and ask him to curl around your inside leg. This is something that you work on away from the barrel pattern, doing lots and lots of circle work. You want it so that when you lift your inside rein and give him a tiny bit of inside leg pressure (slightly forward) that he just wraps his body around the barrel.
FINISH YOUR TURNS. You let him drift on almost every single one. This is a horrible habit to create, and to fix. When you are practicing (at the trot ... right?) make sure to circle each barrel at least twice before moving on, to make him understand he's got to finish that barrel before going onto the next one.
Same exact issues on the second barrel. No bend going into the turn, and he comes out wide.
Third barrel was worse, because he started the turn too soon, cut off the backside of the turn, and thus had to leave the barrel extremely wide to avoid tipping it over. This is where it is vitally important to keep your inside leg on him, keep him bent, and don't allow him to start the turn until your knee is AT the barrel or slightly past it.
No, it's not the most horrible barrel run I've ever seen. Not by a long shot.
But allowing him to drift out of the turns now is going to be a big problem later if you don't fix it now.
And it concerns me at the end of your run (like it did in the beginning of the run) how he's throwing his nose into the air, evading bit pressure, and ignoring the bit. I don't remember --> Did you say you have had his teeth checked? If he's got a tooth pain issue, it could be causing a bit issue.
If he doesn't have any mouth pain, then you've got to get him softer in the bit. And that's where a trainer would help you immensely because timing is so critical in getting a horse softer in the bit.