People close to Prince Andrew said he wanted to address the issues head-on and did so with "honesty and humility".
It came after the prince's interview with BBC Newsnight on Saturday was described as a "car crash".
In the interview, the prince denied having sex with a then 17-year-old girl - Virginia Giuffre.
Former Buckingham Palace press officer Dickie Arbiter described the interview as "excruciating".
The BBC's royal correspondent Jonny Dymond said the prince was "very damaged" by the interview and the opportunity to clear his name had "failed, badly".
For several months the Duke of York had been facing questions over his ties to Epstein - an American financier who, at the age of 66, took his own life while awaiting trial on sex-trafficking charges.
Prince Andrew "categorically" denied having any sexual contact with Virginia Giuffre known at the time as Virginia Roberts.
The first occasion, she said, took place when she was aged 17.
A lawyer for some of Epstein's alleged victims urged the prince to talk under oath to the US authorities.
Asked about the prince's decision to be interviewed by BBC Newsnight's Emily Maitlis, Mr Arbiter said he thought many questions would be asked in Buckingham Palace.
He said: "They will be wondering: Was this the right decision? Was the right decision made? Who made the decision to put him on? Did he make it himself or did he seek advice within the Palace?
"My guess is that he bulldozed his way in and decided he was going to do it himself without any advice.
"Any sensible-thinking person in the PR business would have thrown their hands up in horror at the very suggestion that he puts himself up in front of a television camera to explain away his actions and his friendship with Jeffrey Epstein."
He added that the interview was "not so much a car crash but an articulated lorry crash".
Mr Arbiter said he believed the interview would have an impact on the Duke of York's relationships with various charities.
Ahead of Saturday's interview, Prince Andrew's ex-wife Sarah Ferguson wrote of her support for him on social media.
She said: "I am deeply supportive and proud of this giant of a principled man, [who] dares to put his shoulder to the wind and stands firm with his sense of honour and truth."
Former BBC royal correspondent Jennie Bond said the interview reminded her of one Princess Diana gave to Panorama in 1995 where she "spilled her soul".
Mrs Bond added that Princes Andrew's lack of remorse in his interview was a "glaring hole".
'Speak to FBI'
Gloria Allred, who is representing some of the young women who say they were victims of Epstein said "there is so much truth that is yet to be revealed".
She added: "I would say to Prince Andrew: the charges made by [Virginia Giuffre] against you are very, very serious charges.
"I think the right and honourable thing to do would be for you to say unequivocally 'I will voluntarily speak to the FBI, I know it is the right thing to do, I have nothing to hide'."
In the lengthy interview, which UK viewers can watch in full on BBC iPlayer or on YouTube elsewhere in the world, the duke said that:
-On the date Virginia Giuffre says he had sex with her - 10 March, 2001, he had taken his daughter to Pizza Express in Woking for a party before spending the night at home
-He dismissed claims he was sweating profusely because he had a "peculiar medical condition" meaning he cannot sweat, caused by an overdose of adrenaline in the Falklands War
-He had commissioned investigations into whether a photograph of him with Ms Giuffre had been faked, but they were inconclusive
-Speaking out about his relationship with the financier had become almost "a mental health issue" for him
-He would testify under oath about his relationship with Ms Giuffre if "push came to shove", and his lawyers advised him to
-He was unaware of an arrest warrant against Epstein when he invited the financier to Princess Beatrice's 18th birthday party at Windsor Castle
-He did not regret his friendship with Epstein because of "the opportunities I was given to learn" from him about trade and business
"Car crash" and "disaster" are some of the kinder words that spring to mind about Prince Andrew's misbegotten foray into the long-form interview.
The reaction of the press and commentators is withering. Social media is burning with mockery, ridicule and a fair amount of anger.
To a fair number of people doubtful about the worth of the monarchy, Prince Andrew has emerged as an avatar of all that is wrong with the institution.
There is a reason the Royals don't do 'no-holds-barred' interviews. Unsurprisingly, given that they live in Palaces and have servants, they are somewhat out of touch.
Which is why Prince Andrew spoke of "a straightforward shooting weekend" and appeared to smirk at the idea of going for a pizza in Woking.
Neglecting to even mention the victims of his friend Jeffrey Epstein compounded the impression of a man who entirely fails to grasp the spirit of the times.
Defending his friendship with a convicted child sex offender on the grounds that he had met lots of interesting people because of him suggested a degree of self-absorption that would not survive exposure to the outside world.
Who in his staff thought this interview would be good idea and what does Prince Andrew do next?
He is very damaged. The interview was an opportunity to clear his name and rescue his reputation. It has failed, badly.
We learn something every day, and lots of times it’s that what we learned the day before was wrong.