Embarrassing private matters taken to laundrette (5,5)
I believe the answer is:
dirty linen
If this is the answer I cannot explain how the clue works.
Can you help me to learn more?
(Other definitions for dirty linen that I've seen before include "The intimate secrets" , "of a laundryman" , "personal problems (like the infamous dress?)" , "Better not wash this in public" , "laundry" .)