Girl, not the first, French teacher quietly upset (9)
I believe the answer is:
priscilla
'girl' is the definition.
(woman's name)
'not the first french teacher quietly upset' is the wordplay.
'not the first' means to remove the first letter.
'french' becomes 'gallic' (I've seen this before).
'teacher' becomes 'sir' (way of addressing a male teacher).
'quietly' becomes 'p' (p for piano and quiet).
'upset' says the letters should be written in reverse.
'gallic'+'sir'+'p'='gallicsirp'
'gallicsirp' with its initial letter taken off is 'allicsirp'.
'allicsirp' written backwards gives 'PRISCILLA'.
(Other definitions for priscilla that I've seen before include "Woman" , "Girl" .)