Eliminate small misgiving, by the sound of it (4,3)
I believe the answer is:
weed out
'eliminate' is the definition.
(I've seen this in another clue)
'small misgiving by the sound of it' is the wordplay.
'small' becomes 'wee' (Scottish term for small).
'misgiving' becomes 'doubt' (I've seen this before).
'by the sound of it' shows a homophone (sound like).
'doubt' sounds like 'dout'.
'wee'+'dout'='WEED OUT'
(Other definitions for weed out that I've seen before include "Remove unwanted elements" , "Eliminate" , "Identify and get rid of" .)