The royal family’s “right to reign” has been questioned in the wake of the Duke of York’s civil sex assault case and the alleged cash-for-honours scandal that has engulfed the Prince of Wales’ foundation.

Charles has made his first public appearance since a Metropolitan Police investigation was launched into the allegation, which has prompted commentators to call for the future king to be questioned by detectives.

The prince hosted a Queen’s Anniversary Prizes for higher and further education awards ceremony with the Princess Royal at St James’ Palace, and appeared relaxed following a turbulent few days for the monarchy.

The Duke of York is being urged to explain how he will pay for his legal settlement with accuser Virginia Giuffre, rumoured to be up to £12 million, as he faces the prospect of never returning to full royal duties.

Charles and his former favoured confidant, Michael Fawcett, were formally reported to the Met Police last September when allegations of cash-for-honours first surfaced in newspaper reports.

Mr Fawcett, who has since resigned as chief executive of The Prince’s Foundation, has been accused of promising to help a Saudi billionaire donor receive British citizenship and a knighthood.

Clarence House has said the prince had “no knowledge” of the alleged cash-for-honours.

A source has said about the police investigation: “His Royal Highness is happy to help if asked. He has not been.”

Labour MP Rachael Maskell, who represents York Central and has called on Andrew to give up his title to show “respect” for people living in the city, suggested in the wake of the events this week that the royal family does not have a “right to reign”.

Ms Maskell was asked on Channel 4 News on Wednesday if recent developments opened up a wider debate about the future of the monarchy and replied: “I think it absolutely does. That’s why we have got to look at the situation of power and privilege in our society and the role that it can play.

“And when we’ve seen that it can be used for exploitation, you know, whether it is the questions over charitable funds or whether it’s the question over personal conduct, then of course there has got to be (questions).

“Nobody has a right to reign and I think we’ve got to seriously ask those questions now.”

Joe Little, managing editor of Majesty Magazine, said the Met Police investigation was another blow to the royal family: “Andrew alone has done a huge amount of reputational damage to himself and more importantly the monarchy.

“The Prince’s Foundation investigation has the capacity to do something equally damaging, but it’s now being taken to the next level because it involves the next king.

“And that’s an extremely tricky situation for him to be in and the monarchy to find itself – it’s another shadow on the Platinum Jubilee horizon.”

The awards ceremony was the first time Charles had attended a public event since it was announced last week he had contracted Covid for a second time.

Charles ended self-isolation on Tuesday but the Duchess of Cornwall remains at home after she tested positive for the virus on Monday, and her place at the prizegiving was taken by the Anne.

A source close to the prince said: “He has bounced back well after his second bout.”

The ceremony was held to celebrate 21 universities and colleges for their innovation and excellence that benefit society and saw Dame Sarah Gilbert, co-creator of the Oxford-AstraZeneca vaccine, collect an award on behalf of Oxford University.