The most common and most likely answer to this crossword clue is the 13 letter word COLD-SHOULDER
I've seen this crossword clue in The Times
C
O
L
D
-
S
H
O
U
L
D
E
R
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 = cold-shoulder.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 COLD-SHOULDER mean?
COLD-SHOULDER verb- To disrespect someone, especially by ignoring them.