George Osborne made a mistake in placing cuts of £1.3bn from personal independence payments (PIP) to disabled people in his budget in a move which triggered the resignation of Iain Duncan Smith, Boris Johnson has said.

As the chancellor’s close political allies warned that he will need to rebuild his reputation with Tory MPs after watching his second budget in a row run into trouble, the London mayor hailed the decision to “push” the welfare cuts to one side.

Speaking on ITV’s The Agenda with Tom Bradby, Johnson said: “The government has decided collectively and quite rightly to take the PIP aspect of it [the budget] and try to sort it out. It’s obvious from what’s happened that its admitted that it was a mistake.”

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The London mayor praised the decision by Downing Street and the Treasury, in the hours leading up to the announcement of Duncan Smith’s resignation on Friday night, to shelve the reforms to PIP which are made to disabled people to help with the costs of aids and appliances.

The former work and pensions secretary agreed with the Treasury to reform the system of PIP after a series of court cases led to a growth in the PIP budget by easing the tests in 10 daily activities for disabled people. But Duncan Smith resigned after saying that it was wrong to cut benefits to disabled people in a budget that also introduced tax cuts for wealthy people in the form of a reduction in capital gains tax.

Johnson, who has joined Duncan Smith in campaigning for a British exit from the EU, indicated that his sympathy for the former work and pensions secretary was limited when he distanced him from some of his criticisms of the Treasury.

The London mayor said: “It would have been much better quite frankly if he had stayed in and fought his point of view from within the cabinet ... I have to say where I do part company from some of the criticism is that I don’t think you can reasonably say this government has lost touch with its mission to help all the people in this country or to be a one nation Conservative government and that is what some people are now saying and I totally and utterly reject that.”

Lord Hague, Duncan Smith’s predecessor as Conservative leader, went further and described his resignation as “unequivocally wrong”. In his weekly Telegraph column, the former foreign secretary challenged Duncan Smith’s criticism of the chancellor for cutting taxes at the same time as cutting disability benefits.

Hague wrote: “Depicting the budget as a zero-sum reduction in welfare spending to pay for tax cuts – or juxtaposed with them, as Iain Duncan Smith put it – is simply playing into the hands of a left wing fallacy that would take us back to Gordon Brown’s policies if put into practice.”

The interventions by Hague and Johnson came as friends and allies of the chancellor said in private that Osborne has been damaged by the resignation of Duncan Smith. His suggestion that the chancellor is failing to act as a one nation Tory, by appearing to show he had little interest in non-Tory voters, has raised concerns about Osborne’s political touch.

Allies say that Osborne, who takes great pride in his reputation as the government’s master strategist, is proving increasingly accident-prone. The chancellor was forced to row back on planned cuts to tax credits, outlined in his summer budget, in his autumn statement in November.

Osborne has told friends in private that Duncan Smith had acted unreasonably. They had agreed the changes to PIP payments after tough negotiations around two weeks before the budget. Duncan Smith then reportedly raised no objections after their agreement.

But the chancellor is understood to be aware of the concerns and realises he is entering a difficult period. Osborne thought hard whether he should have responded to an urgent question in the House of Commons by the shadow chancellor John McDonnell. In the end he decided to leave it to David Gauke, the third most senior member of his Treasury team, after deciding it would set an unhelpful precedent for a chancellor to respond to an urgent question.

Osborne was also wary of appearing in the Commons amid concerns in the Treasury that pro-Brexit Tory MPs could have tried to ambush the chancellor. Gauke, a popular figure who is admired for a light but authoritative touch, received a relatively warm reception from Tory MPs.

One old friend thought the chancellor had made a mistake by not appearing in the Commons. “This was his moment to show us a new George and he has missed his chance,” the friend said. “If he wants to be prime minister, he should be able to turn round a moment like this.”

One ally thinks Osborne is in trouble. “This is damaging for George. He has misjudged a budget again. He has lost a lot of support. You can feel it. He will need to rebuild.”

The chancellor’s old friend believes he is experiencing a bumpy period, but fully expects him to bounce back. “This is not terminal for George. Things come and go. There are waves. Better to have bout of scarlet fever now than in 2018.”