A clergyman lost his head for this crime (5)
I believe the answer is:
arson
'this crime' is the definition.
(crime of illegally starting a fire)
'a clergyman lost his head' is the wordplay.
'a clergyman' becomes 'parson' (parson is a kind of clergyman).
'lost his head' means to remove the first letter.
'parson' with its first letter taken off is 'ARSON'.
'for' acts as a link.
(Other definitions for arson that I've seen before include "Malicious crime" , "Fire-lighting crime" , "Crime of fire raising" , "Felonious burning of property" , "Sonar (anag.)" .)