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"his regret" is correct and can be used in written English.
For example: John felt a deep sense of his regret as he tried to apologize to his friend.
Exact(56)
His regret was palpable.
He never gets over his regret.
Mr. Green gave in, to his regret.
(He conveyed his regret to Merkel privately).
And his regret, whatever it was for, was also real.
He expressed his regret in these three cases.
Concluding his evidence, he expressed his regret at what happened.
Much to his regret, Mr Tourre's fame will continue.
Clearly embarrassed, Schofield issued a statement expressing his regret that the stunt had misfired.
Poles felt that that wording implied Polish complicity, and Obama later expressed his "regret".
"He has expressed his regret, and repaid the money wrongly claimed," the report said.
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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