Finished with their drama - they lay it on a bit thick (8)
I believe the answer is:
overplay
'drama they lay it on a bit thick' is the definition.
The definition suggests an adverb but the answer is not.
'finished with their' is the wordplay.
'finished' becomes 'over' (over can mean finished or ended).
'with' means one lot of letters go next to another.
'their' becomes 'play' (I can't explain this - if you can you should believe this answer much more).
'over'+'play'='OVERPLAY'
Can you help me to learn more?
(Other definitions for overplay that I've seen before include "Give excess emphasis to" , "Complicate or exaggerate (something)" , "Embroider" , "Exaggerate the importance of" .)