Distribute a small notice (6)
I believe the answer is:
assign
'distribute' is the definition.
(assigning is a kind of distributing)
'a small notice' is the wordplay.
'small' becomes 's' (abbreviation - e.g. clothes size).
'notice' becomes 'sign' (I've seen this in another clue).
'a'+'s'+'sign'='ASSIGN'
(Other definitions for assign that I've seen before include "Determine" , "Give or entrust to" , "Delegate" , "award" , "Allot or ascribe" .)