So the Apple IIc (at least, the one I have) has a thin gray membrane of sorts that overlays the keystems. It's badly cracked and deteriorated. My intuition is that it serves no immediately valuable purpose and can probably just be disposed of. Though I'd ask just in case there's a reason to keep it or try to replace it.
That iis a dust shield for the original hairpin keyswitch keys. Its purpose is to keep the contacts clean, and provide moderate spill protection, but aye, you can run without it. Mind was also deteriorated, and disintegrated when I removed it into dozens of fragments.