The most common and most likely answer to this crossword clue is the 8 letter word LESSENED
I've seen this crossword clue in The Guardian
L
E
S
S
E
N
E
D
More clues leading to the result LESSENED
Buddy explains!

The straight part
We know that "Cut" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case cut = lessened.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 LESSENED mean?
LESSENED verb- To make less; to diminish; to reduce.
- To become less.
- Having been lessened.