Bird that woman’s cooking (5)
I believe the answer is:
heron
'bird' is the definition.
(I know that heron is a type of bird)
'woman's cooking' is the wordplay.
'woman' becomes 'her' (her refers to a woman).
'cooking' becomes 'on' (I've seen this in other clues).
'her'+'on'='HERON'
'that' is the link.
(Other definitions for heron that I've seen before include "Grey or white long-legged wading bird" , "Large British wading bird" , "Big bird" , "flier" , "Wading bird with long neck and legs" .)