Judge Kimba Wood had run out of patience. Lawyers for Michael Cohen, Donald Trump’s consigliere and legal fixer, had tried every trick in the book to avoid disclosing the name of a mystery Cohen client, who they said was desperate to remain anonymous.

“I’m directing you to disclose the name – now,” said the veteran jurist.

Stephen Ryan, Cohen’s hangdog lead attorney, got to his feet and cleared his throat. “The client’s name that is involved is Sean Hannity,” he said. Gasps filled courtroom 21B at the federal courthouse in lower Manhattan.

Hannity, the bloviating primetime star of Fox News and Trump’s favorite journalist, had excoriated US prosecutors for their raid on Cohen’s office earlier this month. The raid was part of a “an all-hands-on-deck effort to totally malign and, if possible, impeach the president of the United States”.

He failed to mention his relationship with Cohen, something even guests on his own show have since criticized and an omission that has thrust him into a media maelstrom.

So what is Hannity’s relationship with Cohen, a lawyer most famous for paying off women who allegedly had affairs with the president and one of his billionaire backers? According to Hannity it’s “de minimis”. The multimillionaire may have handed him “10 bucks” for some real estate advice but he had never formally engaged him and there were “no third parties” involved, said Hannity. The media had gone “absolutely insane” about nothing, he said.

For now, it appears his employer agrees. After a short pause in which management quizzed Hannity about his dealings with Cohen, Fox announced it was “surprised by the announcement” but gave him its “full support”.

Fox’s support has failed to appease critics inside and outside the Fox hole. Vanity Fair’s Gabriel Sherman, author of a biography of Hannity’s mentor, the Fox News founder Roger Ailes, wrote this week that fellow Fox staffers were horrified by the news, telling him it “violates every rule of journalism”.

But while Trump’s TV ratings (if not his poll numbers) are still in the ascendant, the Fox star may currently be too big to fail.

“Hannity’s closeness with Trump has given him immense power at the network, and he’s not afraid to show it. When he visited Mar-a-Lago earlier this month, Hannity bragged to a guest: ‘I’m the only thing holding this network together’ (Hannity denies saying this),” wrote Sherman.

Other Fox stars have been cautioned for getting too close to Trump. After Politico revealed that Bret Baier, the network’s 6pm news anchor, had played golf with Trump, the network said it had “addressed the matter”. Hannity too has been challenged for appearing in a Trump ad ahead of the election and for planning a broadcast from a Tea Party rally.

For Hannity’s critics, worse still was his promulgation of the conspiracy theory that Democratic operatives were behind the murder of Seth Rich, a Democratic National Committee staffer, killed in what appears to have been a botched robbery. Hannity continued to push the far-out theory even as Rich’s parents begged him to stop.

There is no suggestion that Hannity is in breach of his contract and the terms of his engagement as a commentator, not a reporter, are likely to be different from Baier’s. Fox said: “We do not discuss talent contracts. They are confidential.”

The bald truth may be that in the current climate, Hannity’s omission just doesn’t matter to Fox.

Last week Fox News marked 14 consecutive weeks as the number one cable TV network. Hannity, with an audience of 3 million viewers, is its biggest star after the channel lost Bill O’Reilly to a sex scandal and Megyn Kelly to NBC. But the median age of its viewers is 68 years old, a demographic timebomb common across cable news (CNN’s is 59).

Media executives expect a big shakeup at Fox in the near future but, for now, they believe Hannity is safe.

One source with close ties to both the Trump administration and Fox described Cohen as “a hanger on” who liked to make himself appear more central to the action than he actually was. He said Hannity would probably emerge from this scandal unscathed - as long as his relationship with Cohen was as he has described.

“Michael Cohen is significantly less of a player than he wants you to believe. Hannity says they had discussions about real estate ... He has tons of real estate lawyers,” the source said. “If it’s just Michael Cohen trying to associate himself with Hannity, I don’t think it’s a problem for Hannity,” he said.

But Hannity’s explanation still puzzles those closely following Cohen’s case.

Cohen’s team chose to include Hannity – then still unidentified – in a list of clients whose files he intends to argue are protected by attorney-client privilege. These records, according to Cohen, should not be viewable by the investigators who are trying to prosecute him.

But Hannity has said Cohen wasn’t really his attorney – he just provided some advice on buying property. “I never retained his services, I never received an invoice. I never paid Michael Cohen for legal fees,” Hannity said, soon after his identity was revealed.

So why did Cohen not just leave Hannity off the list altogether? Investigators stumbling upon mentions of Hannity in Cohen’s files would, presumably, have dismissed the documents as irrelevant and his name should have remained secret.

Instead, investigators tipped off the world to the pair’s working relationship. Researchers now scouring Hannity’s output for clues about it unearthed a clip from Hannity’s show about a week before Trump entered office, in which he made a curious statement.

“I was kidding around yesterday with Michael Cohen – $2bn, some guy in Dubai,” Hannity said. “And I said, ‘Can you give him my number?’ I said: ‘I’m interested in that deal myself.’”

Ultimately Hannity’s undoing may be what has made him such a player at Fox: his closeness to Trump.

Hannity gets 15 mentions in Michael Wollf’s exposé of the early days of the Trump presidency, Fire and Fury – only five fewer than the first lady, Melania Trump. The two talk and meet regularly. Hannity had even planned to set up a rival to Fox with Trump if his presidential bid had foundered, according to Wolff. The Fox star “basically has a desk” in the White House, a presidential aide told the Washington Post last week.

Ultimately those ties may spell trouble for Hannity, said David Folkenflik, NPR media correspondent and author of the book Murdoch’s World. He said the Cohen revelation was at present “a data point” but unlikely to change his relationship with Fox. “But Hannity’s proximity to Trump is such that as we learn more about that relationship, it may be unsustainable. Even for Fox News,” he said.