Keir Starmer has admitted No 10 “didn’t get the process right” in handling the government’s controversial welfare bill and says he shoulders the blame.

Looking to repair some of the damage done by Labour’s 11th hour climbdown on the central plank of its welfare changes, Starmer said the government would reflect on its mistakes.

“We didn’t get the process right. Labour MPs are absolutely vested in this,” Starmer told the BBC podcast Political Thinking with Nick Robinson. “It matters to them to get things like this right, and we didn’t get that process right.”

He added: “We didn’t engage in the way that we should have done.”

Starmer said he took responsibility for the chapter, and repeated his support for his besieged chancellor after recriminations over the government’s U-turn appeared to have left Rachel Reeves in tears at prime minister’s questions in the Commons on Wednesday. He said Reeves would be in post for the next election and “many years after”.

An already under-pressure Reeves has been criticised for her political misjudgment in trying to force through cuts in the face of deep backbench unhappiness. As a result of the U-turn, she now has to fill a £5bn shortfall in planned public savings, either with tax rises or by making cuts elsewhere.

Starmer said Reeves’s tears in the Commons were “nothing to do with politics” and she would be chancellor for a “very long time to come”.

The prime minister denied suggestions the chancellor was upset by the fallout over the government’s welfare bill and insisted he and his chancellor remained “in lockstep”.

He said: “That’s absolutely wrong, it’s got nothing to do with politics, nothing to do with what’s happened this week. It was a personal matter for her. I’m not going to intrude on her privacy by talking to you about that. It is a personal matter.”

He insisted the £5bn gap in public finances would not result in an increase in income tax, national insurance or VAT for working people. “That was a manifesto commitment,” he said. “The one thing we didn’t do in the last budget was we didn’t breach that manifesto commitment. We’re not going to breach that manifesto.”

Acknowledging a “tough” few days, Starmer said his party would “come through it stronger”.

“I’m not going to pretend the last few days have been easy, they’ve been tough. I’m the sort of person that then wants to reflect on that, to ask myself what do we need to ensure we don’t get into a situation like that again, and we will go through that process,” he said. “But I also know what we will do and that’s we will come through it stronger.”

On Thursday, the health secretary, Wes Streeting, said people showing kindness to Reeves “is really appreciated”, and told BBC Breakfast she was “here to stay as our chancellor”. He added that Reeeves was “tough”, had the “courage, strength and judgment” to make tough decisions in the interests of the country and would “bounce back”.

The bill passed its second reading with a majority of 75 on Tuesday, after a climbdown from the government in which it shelved plans for deep cuts to personal independence payments (Pip). The row exposed tensions between No 10 and Labour backbenchers and created a huge headache for Reeves.

Ministers said there would be long-lasting implications for the government’s spending priorities after it was forced to abandon the central plank of its welfare changes to prevent a damaging defeat by rebel MPs. The chaos briefly caused turmoil in financial markets, but by Thursday, Starmer’s insistence that he would stand by his chancellor appeared to have calmed the waters, with overnight rates back to where they were at PMQs, and seemingly unmoved by the U-turn.

Asked by Robinson if he had “lost the dressing room” and the support of his own MPs, Starmer replied: “Absolutely not. As soon as we go through the long list of things that we’ve achieved this year, the Labour dressing room, the PLP [parliamentary Labour party], is proud as hell of what we’ve done.

“Their frustration, my frustration, is that sometimes the other stuff, welfare being an example, can obscure us being able to get that out there.”

Starmer said the government was “only just starting”, adding: “This, in a sense, is the toughest year.”