This sounds just like my QH mare. I just bought her last month and after we got a chance to know one another, I discovered she likes to take off for the gate once you ask for the trot. She will pick her head up, shoulder up, and go immediately into the canter if we are near the gate or her pen, or any direction near the way out lol. She bends her body inward to move outward to the gate. I started carrying a dressage whip. Like the previous poster said, make the wrong thing hard, and the right thing easy. When I felt her tense up and pick up speed near “her direction,” I would turn her into a tight circle. If she picked up the canter, I’d smack her (not hard) on the shoulder on the side of the gate, and turn her head away from it. I did circles constantly in front of the gate. If she didn’t want to go in the gate, I’d back her in. Once she went nicely without fussing, the whip was not used, and she learned that if I squeezed the reins, I meant business. So she’s learning that the easiest thing to do is to not throw a fit, which is what I call it.
It has nothing to do with her not liking arena work. She does great when she’s focused, and not focused on the gate. Her previous owner simply never told her “no,” and let her do whatever she wanted. It’s all about relaxation, and teaching what’s right and wrong. I wish you luck! It gets better