The most common and most likely answer to this crossword clue is the 7 letter word MOPTOPS
I've seen this crossword clue in The Guardian
M
O
P
T
O
P
S
Buddy explains!

The straight part
We know that "cuts" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case cuts = moptops.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 MOPTOPS mean?
MOPTOPS noun- A style of haircut popularized by the Beatles