The most common and most likely answer to this crossword clue is the 6 letter word PLEADS
I've seen this crossword clue in The Sunday Telegraph
P
L
E
A
D
S
Buddy explains!

The straight part
We know that "petitions" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case petitions = pleads.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 PLEADS mean?
PLEADS verb- To present (an argument or a plea), especially in a legal case.
- To beg, beseech, or implore.
- To offer by way of excuse.
- To discuss by arguments.