Eliminate tiny suspicion in conversation (4,3)
I believe the answer is:
weed out
'eliminate' is the definition.
(I've seen this before)
'tiny suspicion in conversation' is the wordplay.
'tiny' becomes 'wee' (I've seen this before).
'suspicion' becomes 'doubt' (doubt can mean suspicion or uncertainty).
'in conversation' shows a homophone (sound like).
'doubt' is a homophone of 'dout'.
'wee'+'dout'='WEED OUT'
(Other definitions for weed out that I've seen before include "Identify and get rid of" , "Remove unwanted elements" , "Eliminate" .)