The most common and most likely answer to this crossword clue is the 6 letter word ABBEYS
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More clues leading to the result ABBEYS
Buddy explains!

The straight part
We know that "monasteries" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case monasteries = abbeys.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 ABBEYS mean?
ABBEYS 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.