Background: what a lady may not wish to reveal about it (8)
I believe the answer is:
heritage
'background' is the definition.
(as in someone's national or ethnic background)
'what a lady may not wish to reveal about it' is the wordplay.
'what a lady may not wish to reveal' becomes 'herage' (traditionally it is rude to ask a woman her age).
'about' is an insertion indicator.
'herage' going around 'it' is 'HERITAGE'.
(Other definitions for heritage that I've seen before include "One comes into this" , "Objects and cultural traditions passed down" , "Birth-related condition" , "historical collection?" , "Property or practices passed down" .)