My first thought is, since you say it takes a long time for him to warm up, is that he may be older and stiffer. Do you know how old he is? Older ones take longer to warm up, but sometimes wraps on the lower legs can help.
So the next thought is ways you can relax, and move your legs and hands independently of your body. Make sure you are relaxed, and breathing calmly and slowly (people tend to hold their breath when tense or nervous). This will help the pony relax, and listen better.
Try pretending as if you are riding, at home, on a rocking chair if you have one. Rock the chair with just the movement of your hips. That is like the canter. While rocking, you can pretend to turn and give leg aids.
For the pony, it would be nice if he could warm up just by a good marching walk for the first few minutes. Also can practice stopping and backing up, just to get pony listening.
Can you ask the instructor if you can come a little earlier to warm up pony more before asking for harder work?
When riding, try to keep your elbows by your waist. That way you will have more strength and shorter reins if pony gets strong.
The hands should not be constant pulling or tension, should be able to relax the hand a bit when pony is doing well. Practice squeezing the inside rein, a few times then relax. Move the inside hand a bit forward for a second when pony gives to the pressure. That is his reward.
Also I have found some horses and ponies really like it if the rider talks to them. Saying things like Easy, or Good Boy can help at times too.
A video would help a lot, but I guess that is not possible...