The most common and most likely answer to this crossword clue is the 6 letter word SWEARS
I've seen this crossword clue in Independent
S
W
E
A
R
S
Buddy explains!

The straight part
We know that "Curses" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case curses = swears.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 SWEARS mean?
SWEARS verb- To take an oath, to promise.
- To use offensive, profane, or obscene language.
- A swear word.
- To be lazy; rest for a short while during working hours.