The most important thing is to sit straight, not hold too much onto the leaping horn with the right leg.
The horse does not 'want to' run/hollow/try to fool you/etc. The reactions in the horse are caused by the actions set up by the rider. If the rider keeps light inside flexion they will have a steadier contact, which aids in bit acceptance (roundness). And in turn inside flexion (ie seeing the inside eyelashes) also gives control of the tempo from the outside rein connection which is developed from the previous.
Work in walk, do circles there, get better bit acceptance, then in trot/etc.
(I have showed/won in side saddle in fei dressage as well as the VA hunter circuit including the Washington National/PA national/Devon/etc.)