Cryptic crossword clue
Offending repeatedly, First Division’s into various crimes 10

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

I've seen this crossword clue in Financial Times
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The straight part
We know that "Offending repeatedly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case offending repeatedly = recidivism.
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.

Crossword Answer definitions

RECIDIVISM noun
  1. Committing new offenses after a crime committed in the past.
  2. Chronic repetition of criminal or other antisocial behavior.
  3. (by extension) Returning to a negative behavior after having stopped it for a period of time.

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