Impress daughter with cool music, starting off with the French (6)
I believe the answer is:
dazzle
'impress' is the definition.
(thesaurus)
'daughter with cool music starting off with the french' is the wordplay.
'daughter with' becomes 'd' (I can't justify this - if you can you should give a lot more credence to this answer).
'cool music' becomes 'jazz'.
'starting off' means to remove the first letter (the starting letter is removed).
'with' says to put letters next to each other.
'the french' becomes 'le'.
'jazz' with its first letter taken away is 'azz'.
'd'+'azz'+'le'='DAZZLE'
(Other definitions for dazzle that I've seen before include "glitter" , "Blind with light or skill" , "Blind with brightness" , "Animal's den" , "Blind with light or brilliance" .)