The most common and most likely answer to this crossword clue is the 3 plus 5 plus 3 plus 3 plus 4 letter word THE GREAT AND THE GOOD
I've seen this crossword clue in The Times
T
H
E
G
R
E
A
T
A
N
D
T
H
E
G
O
O
D
Buddy explains!
The straight part
We know that "the elite" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case the elite = the great and the good.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.