Work on transitions between walk and trot and slowly loosen the reins until you are able to work on the buckle. This teaches your horse to work with loose contact and helps you learn how to use your body and seat to control her pace so that when you pick up the reins she goes happily with a light contact and you still have control. You're not going to accomplish this in one ride so be patient and work on a slightly longer rein until you and your horse can work completely on the buckle
There are a few exercises you could do for this. First though, make sure your horse is in the proper bit. I prefer a snaffle for most exercises. It is one of the mildest bits that you can use. Exercise1: Have a person hold the bit in their two hands, while you hold the reins attached to it. Then, have the person gently push, and pull against you. The idea is that you give, while maintaining a light contact with the bit. for your positions sake, imagine a strait line from your elbow, to your hand, to the bit. It takes practice, but soon your horse should begin to trust your hands. Also, when you ride, try not to use your whole arm when you halt or turn. Try using your seat first, then try opening nd closing your hands to signal the horse. Exercise 2: when you post, try to "pop" your hips forward while holding with your lower leg and thighs. Imagine your head and shoulders just gliding along evenly. This helps control your upper body when you ride.
I hope this helps!
To be honest, unless you're in a life or death situation - I was always taught to never pull on their mouth. Work on transitions by using half halts to keep them balanced.
Bend the elbows. Make sure to give, and go with the flow of the horse. A good start is with the walk, as the head moves quite a lot. Your hands should follow your horse's mouth to allow movement at the poll. If the head doesn't move much then you're holding too much. At the trot, make sure to bend and unbend your elbows as you post. The canter will have similar head movements as walk, but not as much.
Also, use your seat more. I have an OTTB that doesn't like her mouth being touched. So I have taught to speed up, slow down, and turn with legs and seat.
You should maybe try finding a new bit, maybe your mare needs a different one suited for her mouth. also when you are riding give small pulses of your hands. Draw reins could also be useful because they prevented my gelding from running off.