My horse is like that but his "sensitivity" applies to whips and crops. So basically I could kick until my leg fell off with him not caring but so much as SHOW him a whip and I'll get a free flying lesson. He was ok with the whip until I actually used it once (as in touched him, not smacked him) and well..... lets just say after the commotion that then ensued, absolutely nobody who was standing there is willing to whap him with a whip and a kid's mother is completely convinced that I have a monster of a horse. She now comes completely unglued and starts screaming if he so much as walks a bit funny.. yeah, literally screaming her head off for our trainer because I put back boots on him so he was marching when I hopped on. She's convinced he's going to kill me and then take out the rest of the world...
So I've doing the "annoy the crap out of him" method which seems exactly like nagging to me but my trainer swears it's not and it's working, so I suppose she really does know what she's talking about.
So basically ask with gently leg pressure. No result, immediately start kicking (more like a medium pressure tap) repetitively at the same length intervals until he speeds up, moves away from my leg or whatever I was originally requesting. I feel really stupid (quite sure I look really stupid too sitting there going kicky, kicky, kicky, kicky) BUT he's now to the point where if I ask and he ignores, the second my heels go slightly out in preparation to start going kicky, kicky, kicky he literally shots forward with an OMG I AM GOING!!! I AM GOING!!
Now the barn's kid pony that takes an act of God to get a decent walk out of.... ask and then SMACK with the whip while booting him one. Helps if you have the trainer chasing with another whip while screaming too! I wouldn't describe him as "sensitive" though, more like riding a rock. Awesome kid pony though!! Zero worry of him ever bolting!