The most common and most likely answer to this crossword clue is the 13 letter word FAULT-FINDING
I've seen this crossword clue in The Guardian
F
A
U
L
T
-
F
I
N
D
I
N
G
Buddy explains!
The straight part
We know that "petty criticism" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case petty criticism = fault-finding.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.