Woman had an outhouse (4)
I believe the answer is:
shed
This clue is a double definition.
'woman had' is the first definition.
(she'd)
'an outhouse' is the second definition.
(thesaurus)
(Other definitions for shed that I've seen before include "tools here, maybe?" , "Hut - throw off - Guardian cryptic crossword setter" , "that may be at bottom of garden?" , "Threw off the hut" , "In which lawnmower is kept" .)