The most common and most likely answer to this crossword clue is the 6 letter word MOPING
I've seen this crossword clue in Financial Times
M
O
P
I
N
G
More clues leading to the result MOPING
Buddy explains!

The straight part
We know that "having sulk" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case having sulk = moping.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 MOPING mean?
MOPING verb- To carry oneself in a depressed, lackadaisical manner; to give oneself up to low spirits; to pout, sulk.
- To make spiritless and stupid.
- The act of one who mopes.