Senior Labour figures have clashed over whether the party is giving enough attention to the concerns of core working-class voters to beat the Tories at the next general election.

Gloria De Piero, the newly appointed shadow justice minister, and former whip Graham Jones have told the Guardian that Labour’s leadership must reach out to the party’s heartlands to form a majority government.

But their concerns have been dismissed by Dennis Skinner who said that fellow Labour MPs should realise that working-class voters will be ready to support Jeremy Corbyn after the “near victory” in the general election.

The row comes as a study by Policy Network found that C2 social grade voters, typically earning between £21,000 and £34,000 a year, made up the key group that has so far proved resistant to Labour’s surge in support.

On Sunday, the party’s deputy leader, Tom Watson, called for a renewed campaign to shore up support among traditional working-class voters.

Labour exceeded expectations in the general election, leaving Theresa May a diminished prime minister overseeing a hung parliament.

However, many Labour majorities were cut in so-called rust-belt seats – former mining towns, market towns and seaside constituencies in northern England and the Midlands. Some MPs say the party has struggled to hold on to seats that are outside major cities and do not have significant student or minority ethnic populations.

Labour scraped a victory in De Piero’s Nottinghamshire constituency of Ashfield with a majority of 441 – a swing of 8.9% against the party.

De Piero, who was reappointed to the shadow frontbench on Monday, said: “Former coalfield areas like the one I represent are called Labour’s heartlands for a reason: they are the beating heart of our movement.

“We exceeded expectations nationally at the election and we now have the chance to build on our success. In order to do that we need to listen to the voters who left us in seats like my own and act on their concerns.

“An alliance of the new voters who flocked to Labour at the 2017 election and our traditional base will be very hard for a Tory party which is in total disarray to beat.”

Jones, who is standing as a candidate on Wednesday to become a representative of backbenchers on Labour’s parliamentary committee, said he would use the position to fight for greater attention of white working-class voters. He wants Corbyn to address perceptions that the leadership is not patriotic and does not support the deployment of nuclear weapons, he said.

“How thick does this party have to be? We have not learnt the lessons from the rise of the BNP [in the 90s]. Our core voters cannot be taken for granted. These are people who have been let down by political elites for decades. They see themselves as being at the back of every queue.

“We have to talk about their concerns – counter-terrorism, nationalism, defence and community, the nuclear deterrent and patriotism,” said Jones, who represents Hyndburn in Lancashire.

Asked if he was concerned that the party could sound as if it was appealing to racist sentiments, Jones said: “It is not about that. There are genuine concerns about wages being undercut. We have to arrived at a point where these communities in England can no longer be ignored. And if we fail to address it, we are finished.”

Skinner, whose majority in the Derbyshire seat of Bolsover was cut from 11,778 to 5,288, said that concerns of De Piero and Jones misread the concerns of working-class voters.

“We ran a socialist campaign to a near victory. These are claims from rightwingers in the party.

“The reason that people did not vote for Corbyn was because they believed the propaganda coming from Theresa May and the Tories that Corbyn could not win. But now they will know that Labour is on the way to victory. I have no doubt we can see a Labour government soon,” he said.

The report from Policy Network, whose president is Lord Mandelson, found that 48% of voters in the C2 income group voted Conservative last month, compared with 33% for Labour. Almost half of the group (47%) believe that Labour has moved further away from its “traditional working-class supporters”, with only 22% believing that Labour has moved closer to them.

An analysis by David Cowling, a polling specialist and the BBC’s former head of research, found that there was a 2% swing to Labour from the Conservatives, which is greater than when the party was defeated in 2015 but just under the swing Neil Kinnock secured in Labour’s 1992 defeat.

In a regional breakdown of the swing to Labour, he found that the party’s vote went down in north-east England and was lower than the national average in the East Midlands, Yorkshire and Humberside and the West Midlands.

Labour needs to win 64 seats to get a majority, more than twice its net gains from the last election.