"I want us," said David Cameron, "to do better". It had, he said, "been a difficult month". He didn't exactly admit, in an interview on the Today programme on Monday, that his Government was an "omnishambles", but when you heard John Humphrys mention the "fuel fiasco", and "pastygate", and the "granny tax", and the mix up over Abu Qatada, you could almost hear him wince.

But there was one thing, he said, that got him out of bed in the morning, and much earlier, apparently, than most people think. "People come up to me in the street," he said, "and say, 'I've worked all my life and paid taxes, and my wife has worked all her life and paid taxes, but we don't feel the country does enough for people like us'." What kept him going, he said, was a "driving passion and vision" to change the country, and "make it much more on the side of the people who do the right thing".

It isn't clear whether this "driving passion" woke him up when he was at Oxford and Eton. It isn't even clear whether it woke him up when he became leader of the Conservative party, or Prime Minister. But it was very clear, one sunny day in June, six weeks after he wasn't quite elected, that people who did "the right thing" were going to start having a nicer time than people who didn't. "We will," said his chancellor, George Osborne, in his first Budget, "introduce maximum limits on housing benefit", of "£400 a week". No family, he said, at the Tory party conference, "should get more from living on benefits than the average family gets from going out to work".

The "average family" that works, though it's usually just the parents, earns about £26,000 a year. The new cap on housing benefits is £21,000 a year. That's quite a lot of money just to pay on rent, but in London it often isn't enough. In London, you don't have to live in a mansion in Chelsea to rack up well over £21,000. In London, you can rack up well over that even if you live in its poorest boroughs, and particularly if you're an immigrant and, like a lot of immigrants, have a very big family.

So perhaps it's not surprising that some London boroughs have been trying to find cheaper accommodation for their tenants elsewhere. Newham Council, for example, has written to 1,200 housing organisations, asking if they can help. It says the gap between market rents and the local housing allowance has become too big. It says it has been "forced to look farther afield for alternative supply". And one of the organisations it has approached, called Brighter Futures, is in Stoke-on-Trent.

Brighter Futures seems to think that 500 families on housing benefit from London isn't the kind of brighter future it wants. The local Labour MP, Tristram Hunt, seems to agree. Stoke-on-Trent, he said: "Doesn't need the difficult-to-house cases" that "London boroughs have had enough of." He looked forward, he said, to welcoming the Olympic flame from Stratford, but not "east London's exiles". The least London boroughs could do, he said, was "look after their poor and needy". Which is probably as near as a Labour MP will ever get to saying that he doesn't want more "poor and needy" people in his backyard.

Tristram Hunt made it sound as though it was quite an easy thing for a London borough to house thousands of families on benefits when there's a massive shortage of social housing, and when you can't pay higher private rents without breaking the law. Grant Shapps, the Housing minister, seems to think so, too. There was, he told the Today programme yesterday, a £190m fund available to cover the costs of making the change. He had, he said, "made the legislation and guidance clear". Rents, he told Sir Robin Wales, the mayor of Newham, were going down. But the Office for National Statistics seems to agree with the mayor that they're going up.

Is this "social cleansing"? Brighter Futures says it is, and so does Boris Johnson. If "social cleansing" means that poor people can't always afford to live in expensive places, then it is. Clearly, it would be much, much better if governments built more affordable housing. Clearly, it would be much, much better if private rents went down. But a government's control over the private rental market is limited, and mass building programmes take time. In the meantime, there's a housing benefit bill that's £21bn a year, and still going up.

Is it fair? Is it fair that some people who have lived in an area for quite a while may have to move? Certainly, it seems very tough. Is it fair that some people can have as many children as they like, and live wherever they like, and have their very high rent paid by people who can't afford to do either? It doesn't really seem to be. That, presumably, is why, in a recent poll, 76 per cent of the public said they were in favour of the benefit cap, and 69 per cent of Labour voters.

Labour, by the way, has said it's also in favour of a benefit cap, but it hasn't said what this should be. Labour seems to think, at least while it's not in power, that you can please the "hard-working families" who pay the £21bn housing benefits bill, and all the families whose rent it pays. It's a hard, but inescapable truth, which David Cameron, for all his "omnishambles", seems to have grasped, that you can't.

c.patterson@independent.co.uk; twitter.com/queenchristina_