Like everyone else has said, ride him two handed, your instructor should understand you can't really correct major issues by one handing. Anytime I need to take corrective measure on my western horses I go back to 2 handing.
Western riding comes from the seat and leg primarily -- waitaminute... so does English riding... Anyways.. you must have control of your horse through your seat and leg primarily; if you have control of the horse's rib, shoulder, and haunches, you have control of the entire horse - the head is just a hood ornament. You push them into contact for English riding, but your turns, tempo, gait, just about everything comes from seat and leg INTO hand. So your horse has to learn how to hold himself together without your rein, and learn to listen to cues given mostly by the seat and leg. The fact that you said "he doesn't like when I only have control with one hand" indicates that perhaps you and your trainer should step back and look at the horse's training as a whole, and get the horse responding more to seat and leg.