Continue to weaken the carpet (6)
I believe the answer is:
wilton
'carpet' is the definition.
(Wilton is an example)
'continue to weaken' is the wordplay.
'continue' becomes 'on' (I've seen this in other clues).
'to' says to put letters next to each other (I've seen this in other clues).
'weaken' becomes 'wilt' (wilting is a kind of weakening).
'on' put after 'wilt' is 'WILTON'.
'the' is the link.
(Other definitions for wilton that I've seen before include "Kind of English woven carpet" , "Type of woven carpet" , "22 [WILTS] carpet business" , "Could be carpet" , "Woven carpet with a velvety pile" .)