The most common and most likely answer to this crossword clue is the 4 letter word PISH
P
I
S
H
Buddy explains!

The straight part
We know that "To express contempt" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case to express contempt = pish.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 PISH mean?
PISH noun- A sibilant noise (e.g. "psshh") made by birders and ornithologists to attract small birds.
- To try to attract birds by making a sibilant noise (e.g. "psshh").
- To express contempt.
- Of poor quality; very bad.
- Expressing disdain.