Refuse to drop British music such as this (6)
I believe the answer is:
garage
'music such as this' is the definition.
(garage is a music genre)
'refuse to drop british' is the wordplay.
'refuse' becomes 'garbage' (synonyms).
'to drop' indicates named letters should be taken away.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'garbage' with 'b' taken away is 'GARAGE'.
(Other definitions for garage that I've seen before include "Building for housing or repairing car" , "Room for car" , "rock music" , "Attachment to the house" , "Source of 4 dn or 16 dn" .)