Mostly cancel insignificant investigation (8)
I believe the answer is:
scrutiny
'investigation' is the definition.
(scrutiny is a kind of investigation)
'mostly cancel insignificant' is the wordplay.
'mostly' means to remove the last letter (most but not all of the word).
'cancel' becomes 'scrub' (scrub can mean to cancel or erase).
'insignificant' becomes 'tiny' (I've seen this in another clue).
'scrub' with its final letter removed is 'scru'.
'scru'+'tiny'='SCRUTINY'
(Other definitions for scrutiny that I've seen before include "City runs (anag.)" , "Intense study" , "A careful look or examination" , "Close survey" , "Critical examination" .)