91' Sovereign XJ40.....
Under and slightly behind, the front bumper at each corner, there is a square-ish plastic cover of some type... Maybe they cover the computer modules in each corner, not sure.

But the one on the passenger side is hanging down, it has broken off from the 2 screws that hold the front of it in place. I've strapped it up with zip ties for the time being.

What are these pieces called and where can I get them?

 

Maybe you are referring to the 'undertray' panels on either side??
BEC1236 and BEC1237

I think I actually have a few (two?) NEW plastic panels somewhere. They are both Right or Left, I don't have a pair. The right and left are mirror image parts with different part numbers.
I have no idea if they are still available from the dealer. I sold my 1992 XJ40 years ago so I should get all my spares together and get rid of them to new home??

They also use 'special' brass/rubber self sealing fasteners listed as NUT, WASHER, SCREW (#10, #11, #12) as well as self tapping screws and 'Locut Nut' (BBC3242) in the parts catalogue.
Good luck with your search!! A dealer might still have some??

bob