Permit that lady to adopt a kid (7)
I believe the answer is:
leather
'kid' is the definition.
(I've seen this before)
'permit that lady to adopt a' is the wordplay.
'permit' becomes 'let' (synonyms).
'that lady' becomes 'her' ('her' refers to a woman).
'to adopt' is an insertion indicator.
'let'+'her'='lether'
'lether' enclosing 'a' is 'LEATHER'.
(Other definitions for leather that I've seen before include "A tanner produces it from hide" , "Thrash" , "Treated hide" , "Flog; hide" , "Dressed hide" .)