Clergyman loses face giving offence (5)
I believe the answer is:
arson
'offence' is the definition.
(crime of illegally starting a fire)
'clergyman loses face' is the wordplay.
'clergyman' becomes 'parson' (parson is a kind of clergyman).
'loses face' means to remove the first letter (the 'face' of the word is removed).
'parson' with its first letter removed is 'ARSON'.
'giving' is the link.
(Other definitions for arson that I've seen before include "Pyromaniac's crime" , "Fire requiring forensic study" , "A burning crime" , "Iniquitous behaviour" , "It may get you in court" .)