The most common and most likely answer to this crossword clue is the 5 plus 3 plus 5 letter word UNDER THE TABLE
I've seen this crossword clue in The Times
U
N
D
E
R
T
H
E
T
A
B
L
E
Buddy explains!

The straight part
We know that "Secret" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case secret = under the table.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 UNDER THE TABLE mean?
UNDER THE TABLE adjective- Alternative spelling of under the table; in attributive use.