Rear Wheel Spindle Cover

It may pass through the axle - can you post a pic of whats on the other side of the bike .

 

Is that little button in the centre spring loaded, may be a device that grips inside the axle?
And can you clean that bloody wheel rim please

 

Looks like an oil spot to me. If OP posts a pic to show more, we can advise betterer ...

 

It clearly shows how the right side is held on but no clue to the sprocket side, and if you were trying to sell your wares wouldn’t you show them on a clean wheel FFS

 

It could be screwed on using what little thread is available
Try getting a good grip and turn it counter clockwise just like it is a big nut

 

And I’ve got the tea to get ready for Mrs YM’s imminent return from work.

 

My mate had something like this on his speedy, some aftermarket crap. It's was just pushed in, pain to get out.
This looks similar, put a bar in from other side and hit it out, his was very tight if I remember .

 

Plumbers wrench maybe, put some tape on the cap if you want to protect it (presonally I'd throw it in the bin)