Fast food outlet initially lacking music (3,3)
I believe the answer is:
hip hop
'music' is the definition.
(genre of music)
'fast food outlet initially lacking' is the wordplay.
'fast food' becomes 'hip' (I can't explain this - if you can you should believe this answer much more).
'outlet' becomes 'shop' (shop is a kind of outlet).
'initially lacking' means to remove the first letter.
'shop' with its initial letter removed is 'hop'.
'hip'+'hop'='HIP-HOP'
(Other definitions for hip hop that I've seen before include "Kind of pop music" , "Rap music" , "Genre of 1980s and '90s African-American music" .)