Parish priest, losing head, may give serious offence (5)
I believe the answer is:
arson
'serious offence' is the definition.
(arson is a crime)
'parish priest losing head' is the wordplay.
'parish priest' becomes 'parson'.
'losing head' means to remove the first letter.
'parson' with its initial letter taken away is 'ARSON'.
'may give' acts as a link.
(Other definitions for arson that I've seen before include "burning problem" , "Crime of fire-setting (5)" , "Fire-lighting crime" , "Fire-raiser's crime" , "Sonar (anag.)" .)