Aims to pursue vague business left unfinished (5,4)
I believe the answer is:
loose ends
'business left unfinished' is the definition.
I can't judge whether this definition defines the answer.
'aims to pursue vague' is the wordplay.
'aims' becomes 'ends' (I've seen this before).
'to pursue' means one lot of letters go next to another.
'vague' becomes 'loose' (I've seen this before).
'ends' after 'loose' is 'LOOSE ENDS'.
Can you help me to learn more?
(Other definitions for loose ends that I've seen before include "details yet to be explained" , "Unfinished business" , "items to be settled" .)