Stick with cold response one's covered up with lip (9)
I believe the answer is:
criticism
'stick' is the definition.
(I've seen this before)
'cold response one's covered up with lip' is the wordplay.
'cold' becomes 'c' (eg on taps).
'response' becomes 'tic' (I have seen 'Involuntary response ' mean 'tic' so perhaps 'response' could also mean 'tic').
'one's' becomes 'I's' (Roman numeral).
'covered up with' is an insertion indicator.
'lip' becomes 'rim' (synonyms).
'tic'+'is'='ticis'
'ticis' placed into 'rim' is 'riticism'.
'c'+'riticism'='CRITICISM'
'with' is the link.
(Other definitions for criticism that I've seen before include "Unfavourable judgment" , "Fault-finding, censure" , "Analysis and judgment of artistic work" , "Stick" , "Evaluation of a work of literature" .)