The most common and most likely answer to this crossword clue is the 4 letter word WHEW
I've seen this crossword clue in Independent
W
H
E
W
More clues leading to the result WHEW
Buddy explains!

The straight part
We know that "Expression of relief" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case expression of relief = whew.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 WHEW mean?
WHEW verb- To whistle with a shrill pipe, like a plover.
- An expressive sound made indicating the release of one's inner tension; the release of breath; an expression of relief.
- An expression of amazement or surprise.
- To bustle about.