Something needs to be used on top to make it have a firm contact.
OP: The plastic cover is there to make sure that the ribbon cable connects properly with the pins, with it damaged it won't provide full firm connection to the LCD cable.
I think we call these ZIF connectors (the one with the clip and ribbon connector slotting in) (Mostly HTC uses these).
FPC connectors are front panel connectors that have ribbon cables that plug in (The ones you see in iPhones).
More info here from this quote and link:
Cut to size a hard piece of piece of plastic that will go across the cable where the contacts are. Heat up your hot glue gun. Hold down firmly the piece of plastic that you cut,across the contact area with a very thin nail or needle. Apply glue let it harden. Pull the needle out.
I got it to work reliably (the first time) without glue. I found some very thin but sturdy plastic (like some products are packaged in at the store) and cut a small rectangle of it to match the width of the cable exactly. Then, I matched it up at the end of the cable (near the contacts) and shoved both of them in the socket (with the broken clip). With the extra thickness of the plastic on top of the cable, it prevents the cable from sliding out. And you can pull it out, too, so I guess that makes it "semi-permanent"...
Source: https://www.howtogeek.com/forum/topic/broken-zif-socket