Cucumber grown originally by that woman's family (7)
I believe the answer is:
gherkin
'cucumber' is the definition.
(gherkin is a kind of cucumber)
'grown originally by that woman's family' is the wordplay.
'originally' suggests taking the first letters.
'by' is a charade indicator (letters next to each other).
'that woman's' becomes 'her' (I've seen this before).
'family' becomes 'kin' ('kin' can be a synonym of 'family').
The initial letter of 'grown' is 'g'.
'g'+'her'+'kin'='GHERKIN'
(Other definitions for gherkin that I've seen before include "Small cucumber, usually pickled" , "Cucumber - London skyscraper" , "often pickled" , "Type of small cucumber" , "Vegetable of the cucumber family" .)