The most common and most likely answer to this crossword clue is the 6 letter word VOTARY
I've seen this crossword clue in The Sunday Telegraph
V
O
T
A
R
Y
Buddy explains!

The straight part
We know that "monk, say?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case monk, say? = votary.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 VOTARY mean?
VOTARY noun- A person, such as a monk or nun, who lives a religious life according to vows they have made
- A devotee of a particular religion or cult
- A devout or zealous worshipper
- Someone who is devoted to a particular pursuit etc; an enthusiast.
- Consecrated by a vow or promise; consequent on a vow; devoted; promised.