Quick look as soon as finished (4-4)
I believe the answer is:
once-over
'quick look' is the definition.
(as in giving something the once-over)
'as soon as finished' is the wordplay.
'as soon as' becomes 'once' (I've seen this before).
'finished' becomes 'over' (similar in meaning).
'once'+'over'='ONCE-OVER'
(Other definitions for once-over that I've seen before include "Quick appraisal" , "Quick survey" , "Quick inspection" , "Brief check" , "Appraising glance" .)