Cryptic crossword clue
Cats and mice running around, besetting one’s simple existence 10

The most common and most likely answer to this crossword clue is the 10 letter word ASCETICISM

I've seen this crossword clue in The Times
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Buddy explains!

The straight part
We know that "simple existence" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case simple existence = asceticism.
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 ASCETICISM mean?

ASCETICISM noun
  1. The principles and practices of an ascetic; extreme self-denial and austerity.

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