When you overlap part of the mesh when using puppet warp the mesh erases pixels below it with pixels that are empty. Empty pixels in the mesh should be transparent so that you can see the overlapped pixels below it and only the pixels being covered by other pixels should not be visible.
It looks like you have white pixels selected around your shape. If you're cutting pixels out of a while background the white pixels will show up like your example. If your shape is floating, and not cut out of white background you won't have this issue.
The shape was on its own layer, the white pixels were transparent, but are wiping out what's under them and revealing the background below. Here is another screenshot showing it with the background turned off. I am also curious on how you got your mesh so tight to your object, when ever I shrink the expansion, it starts encroaching too far. On a side note, I'm also using nearest neighbor, and the pen tool.