The King, alongside Camilla, Queen Consort, was attending a reception at the castle where he met with members of the Northern Ireland Assembly.
He paid tribute to Queen Elizabeth II's "shining example" on the visit.
But, during a book-signing ceremony, the King realised he had signed the wrong date in footage that has since gone viral.
In the clip, the King can be seen complaining about the pen, before the Queen Consort joined him to say it was "leaking everywhere".
He can be heard saying: "I can't bear this bloody thing!"
It came as the King, responding at Hillsborough Castle to a message of condolence on behalf of the people of Northern Ireland from Stormont Assembly speaker Alex Maskey, spoke poignantly about the Queen.
He said: “Through all those years, she never ceased to pray for the best of times for this place and for its people, whose stories she knew, whose sorrows our family had felt, and for whom she had a great affection and regard.
“My mother felt deeply, I know, the significance of the role she herself played in bringing together those whom history had separated, and in extending a hand to make possible the healing of long-held hurts.”
With his Queen Consort, Prime Minister Liz Truss and significant figures from Northern Ireland watching, the King said about the late Queen: “Now, with that shining example before me, and with God’s help, I take up my new duties resolved to seek the welfare of all the inhabitants of Northern Ireland.”
Queen Elizabeth II's efforts to foster peace on the island of Ireland were later highlighted by John McDowell, Anglican Archbishop of Armagh and head of the Church of Ireland.
Speaking at Belfast’s St Anne’s Cathedral during a service of reflection, he told the congregation: “The word which I think will be most associated with Queen Elizabeth and Ireland, north and south, is reconciliation.
“It is a great New Testament word and it is a great civic word, and it is a hard word.
“So hard in the religious sense that it was beyond the power of humanity to achieve, and God himself had to give it to us as a gift in his son.
“And as a disciple of Jesus Christ, Queen Elizabeth followed where Jesus led, as women often have in the elusive and unfinished work of reconciliation here in Ireland.”
The Queen Consort did her best to help the monarch
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King Charles III using the pen on Tuesday
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