The plastic pie plate is "spoke protector" aka dork disk, which just goes to show what many more experienced riders think of it. It's purpose is to protect the spokes if the chain is overshifted beyond the largest sprocket, and on many bikes, is also to keep the RD cage from snagging in the spokes. Putting the RD into the spokes is an expensive proposition, so this added measure of protection is worth the added weight or name calling.

Better bikes usually lack spoke protectors because their owners can be relied on to keep their bikes properly adjusted, and to an extent, better derailleurs hold adjustment better.

Keep in mind, that it's not only a matter of the derailleur adjustments. If the bike falls on it's right side, the hanger on which the RD can get bent inward, throwing all adjustments off in that direction, and causing an overshift next time you shift to low.

So that's why it's there, and it's your call whether you want one or not -- I'm 100% neutral, and don't use one, and don't advise against them either.

Since it's broken there are two ways to remove it. If the metal part where the plastic was affixed isn't also bent and causing problems, you can simply cut the plastic and discard it. Or you can remove it properly by removing the cassette and slipping the entire disc off, then remounting the cassette. In some cases you may need a spacer to replace the thickness of the disc, but not always.