You need to stretch down more and support more with your leg. If you get better support with your legs, he won't trot in the middle of your course, and you'll be able to make turns easier. You should be able to get rid of your reins and still stay secure in your seat. To help this, try tying up your reins and trotting and cantering around with your hands on your head, or held out to your sides. Get a friend to put you on a longe line so you don't have to worry about steering (make sure your friend knows what they're doing though, lol) Also try riding without stirrups.
You're totally tugging on his mouth over the jumps. If you keep tugging, this could teach him that jumps=tugging=jumping hurts. Which is never a good association. Put your hands forward over the fence. Practice grabbing mane over the fences so you don't tug on his mouth. Once you have the feel of that, you can let go of the mane. But you should jump while holding mane for a while (definitely more than one day) before letting go, as silly as it may feel. It's just practice. Advanced riders on green horses do the same thing sometimes. It's just not fair to him to catch him in the mouth all the time. Look up "jumping position" on wikipedia; they have a bunch of stuff on releases that might be good to read.
Lean forward, forward, forward. You're sitting behind the vertical sometimes. If you want to do hunters, that's not good.
Lift your hands up and forward. Carry them, don't let them just sit there in your lap. If I had to pick, this would be the first thing I want you to focus on in general. Shorten your reins. This will help him not bury himself at the foot of these fences so much.
Don't pump with your seat. It's not doing anything. This is one of my bad habits too, always has been. I don't know if you can feel it or not... just picture a string from the top of your helmet, pulling you up so you can't move back and forth with your body. Or something. That's just what I've always used.
Don't just run him at the jump. This is a green horse, and he needs more help. He's being a saint and going over everything, but these jumps are low. If you want to jump any higher, you need to help him out a bit more. See the distance you need and lengthen or shorten his stride to get to that distance. Set a canter pole before a fence to get the feel for the right distance and/or play the stride game. Start by counting one stride before a fence. When you get that right, come around and count two strides out from a fence. Once you get that, come around and count three, and so on and so forth. Do you know what I mean? Sorry, it's kind of hard to explain. It helped me a lot, though.
DO NOT TROT. In the hunter ring, trotting in the middle of a course is basically a quick and easy way to get last place. In jumpers you can go around a course trotting in the middle. In hunters that won't fly. And I know it's hard. I have a green horse too, who doesn't like to stay in the canter without a lot of work on my part either.
You're hunching your back over the fences. Keep that back straight or arched. Also, on a few fences you're coming out of your jumping position too soon which I'm sure is hurting his back. So just stay up a bit longer.
Anyway, sorry, I know that's a lot of stuff to digest. You've got a lovely little horse there and I'm sure you two will be great together. But everything is a two-sided effort. If you work on your eq, you will feel a positive response from him. There's a reason for eq! If you work on it, I think you two could do hunters. But remember, hunters is all about style. And even if its supposed to be judged on your horse, your horse's looks and jumps are influenced by how you ride. And with a green horse, you need to ride twice as well as everybody else to do the same job. And it's all about practice. Practice, practice, practice, and a ton of concentration.
You're not a bad rider-you've got the basics, and I'm sure the show was stressing you out. I'm making a lot of things sound more awful than they actually are. I've had most of these problems at sometime or another myself. Sorry to be so picky! But if you work on these things a bit I think you can do the hunters. Hunters is about being slow, even, pretty, and organized. You're fine on the speed, your horse is quite cute, and so now you just need to be a bit more organized and crisp. If you do that, you could be winning ribbons! Good luck!
Btw this is a great reference sheet type thing. It's worth a read. Horseback Riding - Position On The Horse
Also watch videos of people with good eq, like this. It helps. :)