Short assignment (6)
I believe the answer is:
errand
'short assignment' is the definition.
(I have seen 'Short assignment ' mean 'errand' so perhaps 'assignment' could also mean 'errand'; 'short' suggests an abbreviation)
This is all the clue.
(Other definitions for errand that I've seen before include "Gone up" , "run one" , "Commission, message" , "Trip for small task" , "Mission (usually run!)" .)