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The phrase "acknowledge one mistake" is correct and usable in written English.
You can use it when referring to the act of admitting or recognizing a single error or fault.
Example: "In his apology, he chose to acknowledge one mistake that had caused significant confusion among the team."
Alternatives: "admit a single error" or "recognize one fault".
Exact(1)
But he did acknowledge one mistake.
Similar(59)
The humility to acknowledge one's own mistakes and shortcomings.
An Army investigation concluded in August that Private Beauchamp's reports were false, but aside from acknowledging one factual mistake, the soldier continued to insist they were true.
Acknowledging one's mistakes is generally considered a virtue.
He acknowledges making one mistake.
In several hours of conversation Pugachev acknowledges just one mistake.
In the interview, Mr. Gore acknowledged one incident recently in which he uttered a false statement, although he preferred to call it a "mistake".
The Israeli Army chief of staff, Lt. Gen. Shaul Mofaz, subsequently told a parliamentary committee that the raid had not attained the "intended results," that it would be investigated and that Israel would acknowledge a mistake if one had been made.
One mistake, one goal.
One mistake too many.
When the film was shown at the New York Film Festival this year, no one mistook it for a technical marvel, but critics acknowledged its enormous power.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com