Something left out of old assignment (8)
I believe the answer is:
omission
'something left out' is the definition.
(I've seen this before)
'old assignment' is the wordplay.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'assignment' becomes 'mission' (mission is a kind of assignment).
'o'+'mission'='OMISSION'
'of' acts as a link.
(Other definitions for omission that I've seen before include "Failure to include" , "Gap" , "Mistake resulting from neglect" , "Act of leaving out" , "Exclusion" .)