I can speak from experiance- but stay calm, as horses can feel your tension and nerves which will make them excitable... get to know your horse really well and talk to him calmly. Sometimes, if you canter him at the end of a lesson when he's a bit more worn out than the begining- do it inside first in the arena then go for longer periods of time........

The big tip is stay calm and hopefully he won't be as edgy and keep your position!!!

I hate to say this but you need to do more work on your position schooling without stirrups at walk and trot, then canter, also going over jumps with no stirrups and arms folded. This sounds brutal and it is I was taught to ride at a riding school in the 1960's taught by a Lt.Col.Leech ex-Indian Army Cavalry. Taught to stay in the saddle but there is a down side going down with the horse still in the saddle, thats happened to me too many times with many horses in 44 years of riding and my own horse for 21 years. I'm having problems with my new horse an ex-racehorse rescue who reared with a eggbutt jointed snaffle but when changed as instructed to a French Link eggbutt snaffle it works well. Just keep at it everything will come good.

As it is a few weeks since your query you may be fine by now but if not - then why not allow your instructor or some other trusted person attach a lunge line to your mare's bit?   At first just canter her on the lunge without anyone on board and then you sit up and allow yourself to have a trot first and then when comfortableand secure that your friend/instructor has the other end of the lunge line, request a canter.   If you hook a finger under the neckstrap of the martingale you will also steady yourself and be able to sit quietly.  You will be fine!  Dont forget to visualize yourself cantering happily and in a relaxed position as this will help your mental state.  Also never forget the old saying that we have here in Ireland "what is in the brain - goes down the rein" !!