To add to Spinebot's answer, at least for your specific example, you'd usually do an eyeball not by using a single image as a mesh and moving the pupil pixels, but by having transparent eye holes, then place the pupil behind it -- no weights.
You can't always place an image on top to hide things. In that case you can use clipping to hide them. While this does have a hard edge, sometimes that can be overcome by placing an image on top of the clipping edge.
More advanced clipping has great uses, but also some technological challenges. We are always working on improving Spine, though new clipping features aren't on our short term radar.