Thirty seconds after taking his seat in the conference room of Nottingham Trent University, which is a makeshift base camp on the Labour leader hustings trail, leadership contender Owen Smith is taking credit for defeating the Tory government’s £4.4bn cut to personal independence payments (PIP) for disabled people earlier this year.

“I’ll be really blunt,” says Smith. “It was the Department for Work and Pensions (DWP) shadow team – me, Angela Rayner, Debbie Abrahams, who fought that campaign. Probably the low point of the Tory government. It was just sneaky, wasn’t it?”

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At this point in Labour’s leadership battle between Jeremy Corbyn and Smith, the latter’s account of events speaks to a much bigger question: who can be trusted to lead the party’s fight against the Conservative cuts to benefits and support that disabled people depend on?

In the shift towards an anti-austerity agenda, and a backlash in the membership against the party’s previously meagre opposition to government attacks on benefits and disability rights, this is crucial territory. Corbyn used the launch of his leadership campaign last month to reiterate his support for social security, pledging to defend the party’s “greatest creation” of the welfare state if re-elected. He has been a long-term, vocal critic of cuts to disability benefits and care packages, which crystallised in his voting against the Conservatives’ welfare bill. Crucially, his team have continually taken credit for the high-profile victories over the PIP disability benefit cuts in March as well as tax credit changes although this has been challenged by Labour’s leader in the Lords, Lady Angela Smith.

“You know, I think Jeremy’s right to be proud on these things,” Smith says, pausing. When he took over as shadow work and pensions secretary under Corbyn’s leadership last September, he aimed to distance himself from his predecessor Rachel Reeves who had said that Labour was “not the party of people on benefits”.

But he says of his nine months as a shadow minister that he had only one meeting with Corbyn. “Just one meeting. And he didn’t raise tax credits, universal credit, or the PIP issue,” says Smith. Why does he think that was? “I don’t think Jeremy does detail, to be perfectly honest,” he replies.

Smith says he understands that Corbyn being the only candidate in last year’s leadership contest to vote against the welfare bill was significant for many. Smith abstained but recounts that he was “one of a couple of people” who argued in the shadow cabinet meeting when Harriet Harman was interim leader that the party should vote against the bill. “It was undoubtedly, for lots of people, a watershed moment … But I don’t think anyone can look at my record and say I’ve been anything other than full bore on it,” he says.

Now Smith is focused on putting into action major changes to social security and support for disabled people. Last month, he announced that if elected, he would scrap the DWP, replacing it with a Ministry for Labour and a Department for Social Security. Smith tells me this plan would involve doubling the representation for disabled people: appointing one minister to help those able to work and another, within a separate department, “whose job it is to make sure people are properly supported” through social security.

A key measure, Smith says, would be to rewrite the eligibility rules for PIP, as campaigners have called for, abandoning “rock solid” criteria that don’t reflect the subtlety of people’s conditions, such as whether someone can walk 20 metres. The work capability assessment – the controversial test that determines whether disabled people are “fit to work” – would also be scrapped, and he says he would shift from using private firms to carry out the tests to what he describes as proper assessments in the NHS and social services with a focus on people’s own GPs and specialists. “If you’re being paid by results to get people off benefits, that’s what you’ll do,” he says of the private companies.

This is personal for Smith. His brother has epilepsy and has been receiving employment and support allowance (ESA). “You meet him and you think he’s fine,” he says. “But he finds it so hard to work.”

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Asked if he would consider increasing the ESA rate, he says “Yes, yes. £105 a week is not a lot to live on. Loads of people will be reading this, going to a supermarket and spending £100 on a couple of days’ shopping. People need to reflect on that – we’re a very wealthy country and some people are living on so little.”

This perhaps points to the central challenge facing not only Smith but the Labour party as a whole: how to communicate such policies to an electorate that is increasingly hostile to so-called bloated welfare. Smith suggests two tactics: being unafraid to be completely clear that people who can’t support themselves should be “protected by the collective”, while doing better at showing the experiences of “often invisible” disabled people who need help.

“I think we’ve become less tolerant, angrier, more divided, uglier [as a society],” he says.

For Smith, challenging this means being “a champion for people where it matters”. So what separates him and Corbyn on disability issues? “Ultimately, we can only offer better support if we’re in government,” he says. “That’s the big difference between me and Jeremy. This isn’t about protest … this is about a programme of doing things.”