The most common and most likely answer to this crossword clue is the 7 letter word BLEEPED
I've seen this crossword clue in The Daily Telegraph
B
L
E
E
P
E
D
More clues leading to the result BLEEPED
Buddy explains!

The straight part
We know that "Cut out obscenity" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case cut out obscenity = bleeped.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 BLEEPED mean?
BLEEPED verb- To emit one or more bleeps.
- To edit out inappropriate spoken language in a broadcast by replacing offending words with bleeps.