Red Wall MPs should push for the two-child benefit cap to be lifted rather than a reversal of the winter fuel payment policy, Baroness Harriet Harman has said.

Baroness Harman, the former Labour Party chair, told Sky’s Electoral Dysfunction podcast that this would hand the group a “progressive win” rather than simply “protesting and annoying Sir Keir Starmer” over winter fuel.

Earlier this week, a number of MPs in the Red Wall – Labour’s traditional heartlands in the north of England – reposted a statement on social media in which they said the leadership’s response to the local elections had “fallen on deaf ears”.

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They singled out the cut to the winter fuel allowance as an issue that was raised on the doorstep and urged the government to rethink the policy, arguing doing so “isn’t weak, it takes us to a position of strength”.

Labour’s decision to means test the policy has snatched the benefit away from millions of pensioners.

But Baroness Harman said a better target for the group could be an overhaul of George Osborne’s two-child benefit cap.

The cap, announced in 2015 as part of Lord David Cameron’s austerity measures, means while parents can claim child tax credit or Universal Credit payments for their first and second child, they can’t make claims for any further children they have.

Labour faced pressure to remove the cap in the early months of government, with ministers suggesting in February that they were considering relaxing the limit.

Baroness Harman told Beth Rigby that this could be a sensible pressure point for Red Wall MPs to target.

She said: “It could be that they have a kind of progressive win, and it might not be a bad thing to do in the context of an overall strategy on child poverty.

“Let’s see whether instead of just protesting and annoying Sir Keir Starmer, they can build a bridge to a new progressive set of policies.”

Jo White, the Labour MP for Bassetlaw and a member of the Red Wall group, suggested that her party’s “connection” to a core group of voters “died” with the decision to means test the winter fuel payment for pensioners.

“We need to reset the government,” she told Electoral Dysfunction. “The biggest way to do that is by tackling issues such as winter fuel payments.

“I think we should raise the thresholds so that people perhaps who are paying a higher level of tax are the only people who are exempt from getting it.”

A group of MPs in the Red Wall, thought to number about 40, met on Tuesday night following the fallout of local election results in England, which saw Labour lose the Runcorn by-election and control of Doncaster Council to Reform UK.

Following the results, Sir Keir said “we must deliver that change even more quickly – we must go even further”.

Some Labour MPs believe it amounted to ignoring voters’ concerns.

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One of the MPs who was present at the meeting told Sky News there was “lots of anger at the government’s response to the results”.

“People acknowledged the winter fuel allowance was the main issue for us on the doorstep,” they said.

“There is a lack of vision from this government.”

Another added: “Everyone was furious.”

Downing Street has ruled out a U-turn on means testing the winter fuel payment, following newspaper reports earlier this week that one might be on the cards.