Everyone knows that Clint was the one to capture and spare Natasha from having to kill her, which was his assignment from his SHIELD superiors, but there’s one thing that the rest of the world doesn’t know.
Natasha may have been really young at the time, but so was Clint. He was just over fifteen when he was sent to kill her, and she was thirteen. SHIELD has records of this, but if the information got out to the public, everyone would know that SHIELD did practically the same things the KGB did in order to keep relevant during the Cold War, including child soldiers, brainwashing, and the occasional mind wipe.
And SHIELD can’t let that happen.