Auntie half-heartedly backed naval officer making music (5)
I believe the answer is:
bebop
'music' is the definition.
(bebop is a kind of music)
'auntie half-heartedly backed naval officer' is the wordplay.
'auntie' becomes 'beeb' (I've seen this before).
'half' indicates taking half.
'heartedly' becomes 'b' (I can't justify this - if you can you should give a lot more credence to this answer).
'backed' shows that the letters should be reversed in order.
'naval officer' becomes 'po' (abbreviation for petty officer).
'po' back-to-front is 'op'.
'beeb' cut in half is 'be'.
'be'+'b'+'op'='BEBOP'
'making' is the link.
(Other definitions for bebop that I've seen before include "1940s' jazz characterised by virtuosity" , "1940s' form of modern jazz" , "music played in jazz club" , "Jazz-style dance music of the 1940s" , "Live dance" .)