What happens when Cosmo chooses a facehugger?
What happens when this Cosmo leaves play?
The Facehugger will lose its powers.
"Hug" has already happened so Cosmo doesn't affect that. (The Facehugger will keep its captive.)
"Burst" is now no longer usable while Cosmo is in play.
When Cosmo leaves play, "Burst" will be available again.