Cotton on back of carpet and rug (4)
I believe the answer is:
twig
'cotton on' is the definition.
('twig' can be a synonym of 'cotton on')
'back of carpet and rug' is the wordplay.
'back of' indicates one should take the final letters.
'and' says to put letters next to each other.
'rug' becomes 'wig' ('rug' is an informal term for a wig).
The final letter of 'carpet' is 't'.
't'+'wig'='TWIG'
(Other definitions for twig that I've seen before include "Small branch - divining rod - realise" , "Small woody shoot" , "Small part of tree - suddenly understand" , "Suddenly catch" , "Small piece of wood" .)