A sweetheart full of cold getting recess (6)
I believe the answer is:
alcove
'recess' is the definition.
(alcove is a kind of recess)
'a sweetheart full of cold' is the wordplay.
'sweetheart' becomes 'love' (I've seen this before**).
'full of' is an insertion indicator.
'cold' becomes 'c' (eg on taps).
'love' going around 'c' is 'lcove'.
'a'+'lcove'='ALCOVE'
'getting' is the link.
(Other definitions for alcove that I've seen before include "sheltered spot" , "Small opening - a clove (anag)" , "Small opening off a larger room" , "private part?" , "A small recess or niche" .)