The most common and most likely answer to this crossword clue is the 4 letter word DAMN
I've seen this crossword clue in The Times
D
A
M
N
More clues leading to the result DAMN
Buddy explains!

The straight part
We know that "mild curse" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case mild curse = damn.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 DAMN mean?
DAMN noun- The use of "damn" as a curse.
- A small, negligible quantity, being of little value; a whit or jot.
- The smallest amount of concern or consideration.
- To condemn to hell.
- To condemn; to declare guilty; to doom; to adjudge to punishment.
- To put out of favor; to ruin; to label negatively.
- To condemn as unfit, harmful, invalid, immoral or illegal.
- To curse; put a curse upon.
- To invoke damnation; to curse.
- Generic intensifier. Fucking; bloody.
- Very; extremely.
- Used to express anger, irritation, disappointment, annoyance, contempt or suprise, etc. See also dammit.