That lady would get rid of outbuilding (4)
I believe the answer is:
shed
'rid of outbuilding' is the definition.
'shed' can be an answer for 'outbuilding' (shed is a kind of outbuilding). I'm unsure of the 'rid of' bit.
'that lady would' is the wordplay.
'that lady' becomes 'she' ('she' refers to a woman).
'would' becomes 'd' (I've seen this in other clues).
'she'+'d'='SHED'
'get' is the link.
Can you help me to learn more?
(Other definitions for shed that I've seen before include "Outbuilding for storage" , "Kind of hut - discard it" , "Threw off the hut" , "tools here, maybe?" , "Lose or take off in small building" .)