The most common and most likely answer to this crossword clue is the 4 letter word OUCH
I've seen this crossword clue in Independent
O
U
C
H
More clues leading to the result OUCH
- Contact, heading off jet setting once?
- It hurts, removing penny from sporran
- It’s hurtful if there is no time to keep in contact
- Contactthat was painful!
- Stroke, not the first that hurts
- Head shaved in bed, that’s proved painful!
- Exclamation from Scot pierced by end of skean-dhu that hurts!
- Starts to offer unkind criticismthat hurts
- Topless beefcake poser’s wear that hurt?
- Cry coming from contact lacking time
Buddy explains!

The straight part
We know that "that hurts" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case that hurts = ouch.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 OUCH mean?
OUCH interjection- An expression of one's own physical pain.
- An expression in sympathy at another's pain.
- A reply to an insult seen as savage (frequently one that is tongue-in-cheek or joking).
- An expression of disappointment.
- Expressing surprise at the high price of something.
- A brooch or clasp for fastening a piece of clothing together, especially when valuable or set with jewels.