The most common and most likely answer to this crossword clue is the 5 plus 5 letter word DIRTY LINEN
I've seen this crossword clue in The Daily Telegraph
D
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R
T
Y
L
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N
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N
More clues leading to the result DIRTY LINEN
Buddy explains!
The straight part
We know that "Washing that’s best kept secret" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case washing that’s best kept secret = dirty linen.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.