The most common and most likely answer to this crossword clue is the 7 plus 4 plus 1 plus 4 letter word DOESN’T GIVE A HOOT
I've seen this crossword clue in The Guardian
D
O
E
S
N
â
€
™
T
G
I
V
E
A
H
O
O
T
Buddy explains!
The straight part
We know that "couldn’t care less" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case couldn’t care less = doesn’t give a hoot.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.