Rupert Murdoch, James Murdoch and their former editor Andy Coulson all face embarrassing new allegations of dishonesty and cover-up after the publication of an explosive letter written by the News of the World's disgraced royal correspondent, Clive Goodman.

In the letter, which was written four years ago but published only on Tuesday, Goodman claims that phone hacking was "widely discussed" at editorial meetings at the paper until Coulson himself banned further references to it; that Coulson offered to let him keep his job if he agreed not to implicate the paper in hacking when he came to court; and that his own hacking was carried out with "the full knowledge and support" of other senior journalists, whom he named.

The claims are acutely troubling for the prime minister, David Cameron, who hired Coulson as his media adviser on the basis that he knew nothing about phone hacking. And they confront Rupert and James Murdoch with the humiliating prospect of being recalled to parliament to justify the evidence which they gave last month on the aftermath of Goodman's allegations. In a separate letter, one of the Murdochs' own law firms claim that parts of that evidence were variously "hard to credit", "self-serving" and "inaccurate and misleading".

Goodman's claims also raise serious questions about Rupert Murdoch's close friend and adviser, Les Hinton, who was sent a copy of the letter but failed to pass it to police and who then led a cast of senior Murdoch personnel in telling parliament that they believed Coulson knew nothing about the interception of the voicemail of public figures and that Goodman was the only journalist involved.

The letters from Goodman and from the London law firm Harbottle & Lewis are among a cache of paperwork published by the Commons culture, media and sport select committee. One committee member, the Labour MP Tom Watson, said Goodman's letter was "absolutely devastating". He said: "Clive Goodman's letter is the most significant piece of evidence that has been revealed so far. It completely removes News International's defence. This is one of the largest cover-ups I have seen in my lifetime."

Goodman's letter is dated 2 March 2007, soon after he was released from a four-month prison sentence. It is addressed to News International's director of human resources, Daniel Cloke, and registers his appeal against the decision of Hinton, the company's then chairman, to sack him for gross misconduct after he admitted intercepting the voicemail of three members of the royal household. Goodman lists five grounds for his appeal.

He argues that the decision is perverse because he acted "with the full knowledge and support" of named senior journalists and that payments for the private investigator who assisted him, Glenn Mulcaire, were arranged by another senior journalist. The names of the journalists have been redacted from the published letter at the request of Scotland Yard, who are investigating the affair.

Goodman then claims that other members of staff at the News of the World were also hacking phones. Crucially, he adds: "This practice was widely discussed in the daily editorial conference, until explicit reference to it was banned by the editor." He reveals that the paper continued to consult him on stories even though they knew he was going to plead guilty to phone hacking and that the paper's then lawyer, Tom Crone, knew all the details of the case against him.

In a particularly embarrassing allegation, he adds: "Tom Crone and the editor promised on many occasions that I could come back to a job at the newspaper if I did not implicate the paper or any of its staff in my mitigation plea. I did not, and I expect the paper to honour its promise to me." In the event, Goodman lost his appeal. But the claim that the paper induced him to mislead the court is one that may cause further problems for News International.

Two versions of his letter were provided to the committee. One which was supplied by Harbottle & Lewis has been redacted to remove the names of journalists, at the request of police. The other, which was supplied by News International, has been redacted to remove not only the names but also all references to hacking being discussed in Coulson's editorial meetings and to Coulson's offer to keep Goodman on staff if he agreed not to implicate the paper.

The company also faces a new claim that it misled parliament. In earlier evidence to the select committee, in answer to questions about whether it had bought Goodman's silence, it had said he was paid off with a period of notice plus compensation of no more than £60,000. The new paperwork, however, reveals that Goodman was paid a full year's salary, worth £90,502.08, plus a further £140,000 in compensation as well as £13,000 to cover his lawyer's bill. Watson said: "It's hush money. I think they tried to buy his silence." Murdoch's executives have always denied this.

When Goodman's letter reached News International four years ago, it set off a chain reaction which now threatens embarrassment for Rupert and James Murdoch personally. The company resisted Goodman's appeal, and he requested disclosure of emails sent to and from six named senior journalists on the paper. The company collected 2,500 emails and sent them to Harbottle & Lewis and asked the law firm to examine them.

Harbottle & Lewis then produced a letter, which has previously been published by the select committee in a non-redacted form: "I can confirm that we did not find anything in those emails which appeared to us to be reasonable evidence that Clive Goodman's illegal actions were known about and supported by both or either of Andy Coulson, the editor, and Neil Wallis, the deputy editor, and/or that Ian Edmondson, the news editor, and others were carrying out similar illegal procedures."

In their evidence to the select committee last month, the Murdochs presented this letter as evidence that the company had been given a clean bill of health. However, the Metropolitan police have since said that the emails contained evidence of "alleged payments by corrupt journalists to corrupt police officers". And the former director of public prosecutions, Ken Macdonald, who examined a small sample of the emails, said they contained evidence of indirect hacking, breaches of national security and serious crime.

In a lengthy reply, Harbottle & Lewis say it was never asked to investigate whether crimes generally had been committed at the News of the World but had been instructed only to say whether the emails contained evidence that Goodman had hacked phones with "the full knowledge and support" of the named senior journalists. The law firm reveals that the letter was the result of a detailed negotiation with News International's senior lawyer, Jon Chapman, and it refused to include a line which he suggested, that, having seen a copy of Goodman's letter of 2 March: "We did not find anything that we consider to be directly relevant to the grounds of appeal put forward by him."

In a lengthy criticism of the Murdochs' evidence to the select committee last month, Harbottle & Lewis says it finds it "hard to credit" James Murdoch's repeated claim that News International "rested on" its letter as part of their grounds for believing that Goodman was a "rogue reporter". It says News International's view of the law firm's role is "self-serving" and that Rupert Murdoch's claim that it was hired "to find out what the hell was going on" was "inaccurate and misleading", although it adds that he may have been confused or misinformed about its role.

Harbottle & Lewis writes: "There was absolutely no question of the firm being asked to provide News International with a clean bill of health which it could deploy years later in wholly different contexts for wholly different purposes … The firm was not being asked to provide some sort of 'good conduct certificate' which News International could show to parliament … Nor was it being given a general retainer, as Mr Rupert Murdoch asserted it was, 'to find out what the hell was going on'."

The law firm's challenge to the Murdochs' evidence follows an earlier claim made jointly by the paper's former editor and former lawyer that a different element of James Murdoch's evidence to the committee was "mistaken". He had told the committee that he had paid more than £1m to settle a legal action brought by Gordon Taylor of the Professional Footballers Association without knowing that Taylor's lawyers had obtained an email from a junior reporter to the paper's chief reporter, Neville Thurlbeck, containing 35 transcripts of voicemail messages. Crone and the former editor, Colin Myler, last month challenged this.

In letters published by the committee, the former News of the World lawyer repeats his position. He says this email was "the sole reason" for settling Taylor's case. He says he took it with him to a meeting with James Murdoch in June 2008 when he explained the need to settle: "I have no doubt that I informed Mr Murdoch of its existence, of what it was and where it came from."

Myler, in a separate letter also published on Tuesday, endorses Crone's account. Their evidence raises questions about James Murdoch's failure to tell the police or his shareholders about the evidence of crime contained in the email.

Watson said that both Murdochs should be recalled to the committee to explain their evidence. Hinton, who resigned last month, may join them. Four days after Goodman sent his letter, Hinton gave evidence to the select committee in which he made no reference to any of the allegations contained in the letter, but told MPs: "I believe absolutely that Andy [Coulson] did not have knowledge of what was going on". He added that he had carried out a full, rigorous internal inquiry and that he believed Goodman was the only person involved.

Commenting on the evidence from the select committee, a News International spokesperson said: "News Corporation's board has set up a management and standards committee, chaired by independent chairman Lord Grabiner, which is co-operating fully with the Metropolitan police and is facilitating their investigation into illegal voicemail interception at the News of the World and related issues.

"We recognise the seriousness of materials disclosed to the police and parliament and are committed to working in a constructive and open way with all the relevant authorities."