That woman would cast off (4)
I believe the answer is:
shed
I believe this is a double definition.
'that woman would' is the first definition.
('she'd' - contraction of 'she would')
'cast off' is the second definition.
(synonyms)
(Other definitions for shed that I've seen before include "Hut - throw off - Guardian cryptic crossword setter" , "warehouse" , "Abandon" , "made redundant" , "place for bikes" .)