Losing head, churchman gives offence (5)
I believe the answer is:
arson
'offence' is the definition.
(crime of illegally starting a fire)
'losing head churchman' is the wordplay.
'losing head' means to remove the first letter.
'churchman' becomes 'parson' (I've seen this before).
'parson' with its first letter taken away is 'ARSON'.
'gives' is the link.
(Other definitions for arson that I've seen before include "Incendiary crime" , "criminal fire-raising" , "Roans (anag)" , "Crime involving fire" , "major crime" .)