The most common and most likely answer to this crossword clue is the 4 letter word SNIP
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More clues leading to the result SNIP
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 = snip.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 SNIP mean?
SNIP noun- The act of snipping; cutting a small amount off of something.
- A single cut with scissors, clippers, or similar tool.
- Something acquired for a low price; a bargain.
- A small amount of something; a pinch.
- (definite, the snip) A vasectomy.
- A small or weak person, especially a young one.
- An impertinent or mischievous person.
- A share or portion; a snack.
- A tailor.
- To cut with short sharp actions, as with scissors.
- To reduce the price of a product, to create a snip.
- To break off; to snatch away.
- To circumcise.
- To remove the irrelevant parts of quotations in the reply message.