Very minor actor cut me in Cambridgeshire city (9)
I believe the answer is:
extremely
'very' is the definition.
(I've seen this in another clue)
'minor actor cut me in cambridgeshire city' is the wordplay.
'minor actor' becomes 'extra' (as in film extras).
'cut' means to remove the last letter.
'in' is an insertion indicator.
'cambridgeshire city' becomes 'ely' (Ely is a city in Cambridgeshire).
'extra' with its last letter removed is 'extr'.
'me' placed within 'ely' is 'emely'.
'extr'+'emely'='EXTREMELY'
(Other definitions for extremely that I've seen before include "To a high degree or extent" , "remarkably" .)