Converse in supermodularity and single-crossing
I'm not sure there are enough assumptions here to move forward, in particular I would think you would need some conditions on your function $h$, at the very least. It's pretty easy to show that $g(\cdot,\cdot)$ will have increasing differences, which implies the single-crossing property for $g$ (this can be quickly seen by letting $h$ be identically zero). But I don't see how this is enough, since single-crossing doesn't imply supermodularity. Maybe I'm missing something, though.