Distribute a quiet bit (9)

I believe the answer is:

apportion

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'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'

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(Other definitions for apportion that I've seen before include "Distribute" , "Share out" , "Hand out" , "Give out" .)

I've seen this clue in The Daily Express.

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