Hungarian composer's band accepting a piece of music very well (4,6)
I believe the answer is:
bela bartok
'hungarian composer's' is the definition.
(20th-century Hungarian composer)
'band accepting a piece of music very well' is the wordplay.
'band' becomes 'belt' (I've seen this before).
'accepting' is an insertion indicator (inserted letters are accepted).
'piece of music' becomes 'bar' (bar of music).
'very well' becomes 'OK' (synonyms).
'a'+'bar'='abar'
'belt' enclosing 'abar' is 'belabart'.
'belabart'+'ok'='BELA BARTOK'
(Another definition for bela bartok that I've seen is " composer".)