The most common and most likely answer to this crossword clue is the 4 letter word GSOH
I've seen this crossword clue in The Guardian
G
S
O
H
Buddy explains!
The straight part
We know that "What’s possessed by wag/What’s possessed by wag" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case what’s possessed by wag/what’s possessed by wag = gsoh.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 GSOH mean?
GSOH noun- (personal ads) Initialism of good sense of humor.