...instrument that's reportedly part of robbery haul (4)
I believe the answer is:
lute
'instrument' is the definition.
(I know that Lute is a instrument)
'reportedly part of robbery haul' is the wordplay.
I cannot really see how this works, but
'part' could be 'ut' (ut is a kind of part) and 'ut' is found in the answer.
This explanation may well be incorrect...
'that's' is the link.
Can you help me to learn more?
(Other definitions for lute that I've seen before include "Old guitar-like instrument played by minstrels" , "clay" , "Subject to drugs" , "Old plucked instrument" , "instrument that minstrel played" .)