The most common and most likely answer to this crossword clue is the 8 letter word CLERIHEW
I've seen this crossword clue in The Times
C
L
E
R
I
H
E
W
More clues leading to the result CLERIHEW
- Holy man, finishing early, cut verse
- Minister reduced by hack’s comic verse
- … and another almost cut bit of witty verse
- Humorous bio priest cut and cut
- About rent in arrears, perhaps to one’s ears, a type of clue that’s somewhat new
- Short humorous poem
- Lines of retired setters sign up 50 to support Cuba
- Minister of religion caught out, having cut verse
- Edmund C Bentley, for instance mocked gently; Vivaldi, when cut, cut again
- Form of comic verse member of the clergy abridged, then cut
Buddy explains!

The straight part
We know that "comic verse" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case comic verse = clerihew.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does CLERIHEW mean?
CLERIHEW noun- A humorous rhyme of four lines with the rhyming scheme AABB, usually regarding a person mentioned in the first line.