The most common and most likely answer to this crossword clue is the 7 letter word SOLICIT
I've seen this crossword clue in The Guardian
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More clues leading to the result SOLICIT
Buddy explains!

The straight part
We know that "petition" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case petition = solicit.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 SOLICIT mean?
SOLICIT noun- Solicitation
- To persistently endeavor to obtain an object, or bring about an event.
- To woo; to court.
- To persuade or incite one to commit some act, especially illegal or sexual behavior.
- To offer to perform sexual activity, especially when for a payment.
- To make a petition.
- To disturb or trouble; to harass.
- To urge the claims of; to plead; to act as solicitor for or with reference to.
- To disturb; to disquiet.