The Labour party has called for major changes to universal credit to ensure low-income families hit by the coronavirus lockdown receive sufficient financial support throughout the crisis.

The shadow work and pensions secretary, Jonathan Reynolds, called for the five-week wait for a first universal credit payment to be scrapped and replaced with a cash grant, and the two-child limit on benefit payments to be abandoned.

He also called for the government to scrap rules that prevent anyone with savings of more than £16,000 from obtaining universal credit. He said: “For me the system should be viewed as a form of social insurance. It should be there when you need it. We wouldn’t limit the NHS to anyone who had savings of over £16,000.”

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Reynolds also called for recipients of legacy benefits, including employment and support allowance, to receive the same £20 a week uprating that was awarded last month to recipients of universal credit and tax credits.

Universal credit has become a main focus of state financial support for many families since the lockdown, with more than 1.4 million people signing up for the benefit since mid-March, most having either lost their job or seen their self-employment opportunities shrink.

More than 500,000 of the new claimants had taken up advance loans to see them through the five-week wait for the first payment and would have to soon start repaying this sum. Reynolds said these loans – which would be repaid over two years – should be “recoded as a grant and written off”.

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The first “pandemic” universal credit claimants are expected to receive their first payment next Tuesday. Earlier this week, the director general of universal credit, Neil Couling, insisted they would be paid on time, despite the loss of 20% of Department for Work and Pensions (DWP) staff to self-isolation.

Reynolds paid tribute to frontline DWP staff for delivering unprecedented numbers of new claims in recent weeks. But he said that more had to be done to ensure the benefit system was able to support low-income families struggling with the additional costs of lockdown, including food and energy.

Reynolds said that under the new Labour leader, Sir Keir Starmer, the focus of the party’s welfare policy would be on driving down child poverty. “It’s going to be a big focus. We will want to go into an election with a serious plan for how we will reduce child poverty.”

He said Labour wanted to replace universal credit with a more trusted and “less punitive” system, although he was “not talking about a new IT system”. Part of the problem with universal credit was it was viewed with fear and suspicion by potential claimants. “The brand is too damaged,” he said.

Reynolds, the MP for Stalybridge and Hyde, said he wanted changes to the disability benefits system, of which he had personal experience through his son Jack, 17, who has autism. “At the minute it is clear that the voices of people who go though the disability benefits assessment process are not listened to by the government,” he said.