Funny this topic came up. I was just after having a sore elbow myself for a while, which I realized was due to putting too much of my upper body weight onto the handlebar. My position on the bike needed tweaking, so I lower my saddle a few mm, got into a more crouched position where more of the support for my upper body was coming from my midsection, and viola! No more elbow pain.
I'm now doing regular core exercises with a view to improving my, uh, core.