Cotton on rug after end of needlepoint (4)
I believe the answer is:
twig
'cotton on' is the definition.
(both can mean to realise something)
'rug after end of needlepoint' is the wordplay.
'rug' becomes 'wig' ('rug' is an informal term for a wig).
'after' is a charade indicator (letters next to each other).
'end of' says to take the final letters.
The final letter of 'needlepoint' is 't'.
'wig' put after 't' is 'TWIG'.
(Other definitions for twig that I've seen before include "Small tree shoot" , "bit of stick" , "Small part of a branch - suddenly understand" , "Suddenly catch" , "Small woody shoot" .)