Verse cut by priest left unfinished (8)
I believe the answer is:
clerihew
'verse' is the definition.
(I know that clerihew is a type of verse)
'cut by priest left unfinished' is the wordplay.
'cut' becomes 'hew' (hew can mean to chop or cut).
'by' means one lot of letters go next to another.
'priest' becomes 'cleric' (both can mean a clergyman).
'left unfinished' means to remove the last letter.
'cleric' with its final letter taken off is 'cleri'.
'hew' put after 'cleri' is 'CLERIHEW'.
(Other definitions for clerihew that I've seen before include "two couplets" , "Funny lines" , "Satire of rhymed couplets about a famous person" , "Two-lined comic verse" , "few personal lines" .)