Incomplete chirp that is quaint (4)
I believe the answer is:
twee
'is quaint' is the definition.
(I've seen this before)
'incomplete chirp' is the wordplay.
'incomplete' means to remove the last letter.
'chirp' becomes 'tweet' (tweet is a kind of chirp).
'tweet' with its final letter removed is 'TWEE'.
'that' is the link.
(Other definitions for twee that I've seen before include "Excessively sweet" , "Affectedly dainty or refined" , "Pretty-pretty" , "Affectedly cute" , "Excessively pretty" .)