My hand issue definitely does not come from my core :) I understand that normally that is the case but I am extremely strong through my core, to the point where I get told to DISengage it because it's on too strong and my horse can't move properly! My coach, when I can afford lessons, is a horse trainer and competes in Equestrian Australia official dressage on horses she has trained, so I think she knows what she's talking about :)
The coach who first pointed out the hand issue also brought up the core thing, as well as my former severe lack of lower leg stability, as reasons why my hands are a problem, but when I was still riding under him I worked my butt off, worked my core until I couldn't move for two days after a workout, strengthened my lower leg so that now I can jump 3'7" with a strong and steady lower leg, and thus neither is an issue anymore.
However if you want to come to Australia and give me that lesson you're more than welcome to :)
What my issue is, is that I tend to collapse through my ribcage. My sternum comes down and back, and my shoulders roll forward, and my arms and hands end up braced. This is, in me, not a core strength or stability issue, but rather an upper body issue caused by my horrible posture on foot. My shoulders are both stuffed because of my posture, I get bad headaches on a regular basis, and my neck is always tight and sore. It's a physical problem on my account, one which I believe I can fix, but meanwhile, while I'm in the saddle, I need to concentrate on sternum UP and OUT, which then brings my shoulders back, and allows me to carry my hands properly. It IS something I can work on.