Run with a Left objective (4)
I believe the answer is:
goal
'objective' is the definition.
(goal can mean an objective or purpose)
'run with a left' is the wordplay.
'run' becomes 'go' (both can mean to function or work).
'with' says to put letters next to each other.
'left' becomes 'l' (common abbreviation).
'go'+'a'+'l'='GOAL'
(Other definitions for goal that I've seen before include "could be worth 3 points" , "End" , "posts and bar" , "... destination (at football?)" , "Target (best not one's own!)" .)