DAVID Letterman, the American chat show host, occasionally has a segment on his programme called, "Is this anything?" The drill is this: a screen showing the Manhattan skyline rises to reveal an amateur performer doing anything from executing handstands to dancing the cha-cha with his dog.

The screen goes down, Dave says: "Was that anything?", and the answer is either "nothing" or "something".

The producers didn't quite make that other Dave, the British Prime Minister, jump through hoops when he appeared on the show. It might have been less embarrassing if he had. As it was, it turns out that expensive Eton education Mr Cameron enjoyed didn't buy the answers to such pub trivia questions as what is the translation of Magna Carta and who composed Rule Britannia. One awaits Michael Gove, his Education Secretary, sending in the inspectors.

It has been a busy week for British politicians in the US, between Mr Cameron at the United Nations, and Alex Salmond, the First Minister, casually redefining the terms of the independence debate in an interview with the Los Angeles Times before going off to watch the golf. British politicians adore America – it's big and far away and they think that appearing there makes them look big, too. In reality, it only makes them seem more removed from the everyday concerns of those who pay their wages.

Mr Cameron did OK on Letterman if you were in the market for a half-decent Hugh Grant impersonation (that's Four Weddings Hugh, not Scourge of the Tabloid Press Hugh, or Sunset Boulevard Hugh). His team has been touting the appearance as a historic first for a British Prime Minister. The only person this could possibly impress is Tony Blair, who must have been squirming with jealousy.

In Downing Street's eyes, bagging a slot on an American chat show is up there with attending a basketball game with the President and flying on Air Force One, both of which Mr Cameron has already done. You know the sun has truly set on the British Empire when a Prime Minister is happy to vie with Hollywood starlets and dancing dogs for a sliver of the spotlight.

Such is the frequency of the Letterman show and other late-night chat programmes, there is a fair chance that anyone who has been to New York has attended a recording. Guilty as charged. It all began with being stopped in the street by a preternaturally enthusiastic kid with a clipboard asking such testing questions as: "Can you speak English?" and "Do you like to laugh?"

We were herded to a waiting area outside the studio where the queue rivalled that at the post-Christmas returns desk at M&S. Hours passed, then more waiting inside as the warm-up man did his thing. A lifetime later, Letterman appeared. Was he any good? To paraphrase Ken Dodd on Freud, he could never have played the second house at the Glasgow Empire.

Mr Cameron's US trip has confirmed that he has many modes, from courageous statesman lambasting Russia and China over Syria to chat show guest, and is able to slip easily between them. Most of all, however, it has highlighted his inability to get a grip in a crisis. A crisis such as the one caused by a bicycling Conservative Chief Whip, for example.

There has been a concerted effort in some parts to treat the matter of what Andrew Mitchell did as if it was a mere storm in an etiquette teacup. Man has a bad day, man loses his rag, man apologises, nothing much to worry about. As for the clash between what Mr Mitchell says occurred, and what the police log says, well, plead his defenders, that's just another of those heat of the moment things.

It is anything but, of course. The police log of that evening allegedly records Mr Mitchell – who does admit swearing – saying: "Best you learn your [expletive deleted] case ... you don't run this [expletive deleted] Government. You're [expletive deleted] plebs." Such language might have done for Nixon on the White House tapes, but not, it seems, for Mr Mitchell.

When finally ordered to face the cameras and asked if he had used the word pleb, he replied: "I want to make it absolutely clear that I did not use the words attributed to me." What did he say then? Good question. Maybe one day Mr Mitchell will deign to tell us.

Someone is wrong, either the police or Mr Mitchell, and it is very wrong that Mr Cameron is not trying harder to find out who it is. Having had Mr Mitchell "look him in they eye" and deny he used the P word, the matter is now closed as far as the Prime Minister is concerned. Chap has spoken unto chap.

It is not how others see it, though. Mr Cameron already knows how toxic it is for the Tories to be portrayed as uncaring, overprivileged toffs. Hence why you are more likely to see Sweetie the Edinburgh Zoo panda winning the 3.20 at Kempton Park than that Bullingdon Club photograph. Mr Cameron should have asked his Chief Whip to place, on the record, as the police officer did, what he said. In the absence of any such statement, Mr Mitchell's half-baked apology adds insult to insult.

Mr Cameron and the bright boys and girls around him should ask themselves what Mrs Thatcher would have done. Since she won three elections on the back of positioning the Tories as allies of the aspirational working class, it is quite clear she would have sacked Mr Mitchell pronto. She had a soft spot for ex-public schoolboys, of course, but take their part over that of the bobby outside her door?

As Mr Cameron brings his US trip to a close, he should ask himself what will be the abiding memory of this week in most voters' minds. Will they be applauding his stance on Syria, noting his appearance on Letterman, or condemning his handling of the Mitchell matter?

On the first he is limited in his ability to act. The second is neither here nor there. But the third calls into question his judgment, pure and simple. There can scarcely have been a voter who did not put themselves, or a member of their family, in the position of that police officer, even if only for a second.

In that respect, to use the Letterman test, was the Mitchell incident "anything"? It was, Prime Minister. It was a very big something, and your inability or unwillingness to treat it as such will be remembered long after that Rule Britannia boo-boo.