The most common and most likely answer to this crossword clue is the 5 letter word ABBEY
A
B
B
E
Y
More clues leading to the result ABBEY
- It accommodates men in order “to put muscle on the old governor”
- Religious establishment backed vote for protecting bishops
- A black governor of Westminster, perhaps
- Top Cat pinches Dibble’s seat in church
- Where various leaders of British Empire are buried? Yes
- Where all men are brothers?
- Church and evangelical youth leaders welcome bishops
- Bishops, in old-fashioned agreement, upset church
- French priest overlooking yard in devotional site
- A second letter is read out in church
Buddy explains!

The straight part
We know that "monastery" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case monastery = abbey.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 ABBEY mean?
ABBEY noun- The office or dominion of an abbot or abbess.
- A monastery or society of people, secluded from the world and devoted to religion and celibacy, which is headed by an abbot or abbess; also, the monastic building or buildings.
- The church of a monastery.
- A residence that was previously an abbatial building.