If there's any chance that this compact is an antique and worth some
money, your best bet is to take it to any museum and ask the curator of
the museum how that plastic cover would have been held in place.
Museums do a lot of restoration work, and it's likely that if the
curator doesn't know himself, he'd know who to send you to.
If there's no concern over losing value if it's an antique, I would
suggest you take some ordinary white wood glue, apply it to a piece of
paper (preferably with some printing on it so that you can better judge
transparency) give it time to dry, and see what you think.
White glue is milky white when it's in liquid form, but dries clear.
PS: "White glue is milky white when it's in liquid form, but dries
Which means, of course, that there can't be anything that's
actually white in colour inside the glue when it's in liquid form.
Otherwise, the dried glue would be white in colour as well. Truth is,
white wood glue is white for exactly the same reason that clowds,
snowbanks and the head on a beer are all white in colour. And, come to
think of it, none of these things have anything inside them that's white
in colour either. Post again if you want to know how that works.