Running after Carol after tea (7)
I believe the answer is:
chasing
'running after' is the definition.
(I've seen this before)
'carol after tea' is the wordplay.
'carol' becomes 'sing' (I've seen this in other clues).
'after' says to put letters next to each other.
'tea' becomes 'cha' (cha is a type of tea).
'sing' put after 'cha' is 'CHASING'.
(Other definitions for chasing that I've seen before include "work by engraver" , "Pursuing" , "Running after" .)