Priest engaging company to find bug (1,4)
I believe the answer is:
e coli
'bug' is the definition.
(as in E coli bacteria)
'priest engaging company' is the wordplay.
'priest' becomes 'eli' (biblical figure).
'engaging' means one lot of letters goes inside another.
'company' becomes 'co' (abbreviation).
'eli' placed around 'co' is 'E COLI'.
'to find' is the link.
(Other definitions for e coli that I've seen before include "Bacterium found in the gut" , "Some types of this bacterium are toxic" , "cause of ill-feeling" , "cause of serious complaint" , "Type of bacteria causing food poisoning" .)