Distribute a quiet bit (9)
I believe the answer is:
apportion
'distribute' is the definition.
(apportioning is a kind of distributing)
'a quiet bit' is the wordplay.
'quiet' becomes 'p' (music symbol for 'piano', ie soft or quiet).
'bit' becomes 'portion' (I can't explain this - if you can you should give a lot more credence to this answer).
'a'+'p'+'portion'='APPORTION'
Can you help me to learn more?
(Other definitions for apportion that I've seen before include "Distribute" , "Share out" , "Hand out" , "Give out" .)