The next Labour leader must be able to appeal to the general public rather than just party loyalists, Harriet Harman said as she criticised the process that led to Ed Miliband's election.

The deputy Labour leader said the party had asked itself the wrong questions, deciding "who do we like" rather than "who does the country like" in previous contests.

In a speech setting out how Labour would move on from an election defeat that remained "very raw", Ms Harman stressed that the influence of the unions over the decision on leadership had been altered by the move to a "one person, one vote" electoral system.

Tensions over the future of the party burst into the open over the weekend after Jim Murphy declared he was resigning as Scottish leader and delivered a stinging parting shot at Unite boss Len McCluskey as a political "kiss of death".

Mr McCluskey hit back by arguing that the "arrogance" of Blairites such as Mr Murphy was to blame for the failure on May 7.

He warned the union could consider cutting ties with Labour if the next person in the top job was not the "voice of ordinary people".

Questioned about the row after her speech at Labour's headquarters in London she said: "I don't think there is going to be a break between Unite or any of the unions that are affiliated to the Labour Party.

"We have had a lot of soul searching to do across all parts of our party and we will have robust discussions. But, no, I don't think there is going to be a disaffiliation."

She added: "So far as the structural relationship between the Labour Party and the trade unions is concerned, we have got a big change in how we elect our leader.

"We used to have three blocs in the electoral college - the members of the Labour Party, MPs and the affiliated trade unions. Now it is one person, one vote and everybody only has one vote." That was a "major recasting" of the relationship between the unions and Labour, she added.

In response to Mr Murphy's attack on Mr McCluskey, Ms Harman said: "We have had a bitter defeat, we have had a thumping in Scotland. It would be very surprising if some people didn't, from time to time, express an exasperation and frustration and anger.

"These things will be said but we are going to try and keep cool heads and we are certainly not going to change the process and are going to press on."

A Unite spokesman said: "Len McCluskey said very clearly last week that he does not back Unite disaffiliating from Labour."

The next Labour leader will be elected on September 12, and Ms Harman said the contest should involve a series of televised hustings in areas outside the party's heartlands in an effort to engage with the general public.

"Above all, we will let the public in and elect a leader who can lead not just the party but the whole country," she said.

She paid tribute to Mr Miliband as a "thoroughly decent and principled man", but criticised the election process that led to both his appointment and her own as deputy.

Ms Harman said it was time the party had a "deeper and more fundamental debate about our future" than the elections that led to Gordon Brown and Mr Miliband leading Labour.

She said: "When I stood for the deputy leadership it was cosy in front of people who - like us - love politics and love Labour. Very different from the rest of the country.

"We asked ourselves 'Who do we like?'. That was the wrong question. We should have asked - as we made our choice - 'Who does the country like?'.

She added that "if we had done that", perhaps Labour would have chosen charismatic former home secretary Alan Johnson "rather than me".

Under the new leadership election rules, party members, members of affiliated unions and societies, and a new category of "registered supporters" who pay £3 will all have a vote on Mr Miliband's replacement.

Ms Harman said the party had already recruited more than 30,000 full members since polling day.

She said the next leadership election had to be "more public facing" than in 2010.

"If I think back to the 2010 leadership election, I remember a comradely and well-organised debate. I remember hustings that were packed with party members keen to hear what the various contenders had to say.

"We have to get to the heart of why we lost and making the right decisions about how we win. We should not be afraid of differences. We should thrash them out.

"And nor should we be afraid of letting the public in to see those arguments. Because if there is one thought that should drive the thinking as we elect a new leadership team, it is this: which of them has the best qualities and leadership skills most likely to win over the support of the public?

"Not the politically-obsessed public, the people like us, but the people who most of the time are busy getting on with their lives, not thinking about politics."

She said Labour had begun talks with the broadcasters over "robust, tough, televised hustings which involve the public", with events in areas the party had failed to win.

"We have to go back and ask local people from those areas to be brutally honest about what they think of us and what they want from us," she said.

Asked if there might have been a different answer in 2010 if the question at the leadership election which led to Mr Miliband's victory had been about who the country liked, Ms Harman said: "You only change the process if you think the process needs to be different and better and that is why we are changing it.

"It's a very brave thing that we are doing."

Shadow health secretary Andy Burnham has been consolidating his position as the front runner, receiving the endorsement of shadow work and pensions secretary Rachel Reeves.

Mr Burnham denied he was the union candidate in an interview yesterday, insisting he was offering "change" and "unity".

Mr Burnham and the other candidates - so far Yvette Cooper, Mary Creagh and Liz Kendall - could be joined by shadow education secretary Tristram Hunt, who is set to make a decision this week on whether to throw his hat into the ring.

Ms Harman said the party had to move on from the "New Labour, Old Labour, Blairite, Brownite" divisions.

"These labels are unhelpful in what is a different era," she said.

In a message to the party during the current "dark days", she pointed out that following 1992's election defeat Labour went on to three successive victories under Tony Blair.

"I remind you of what it was like then to raise your hopes that great victory can follow shocking defeat. But also to remind you that when we are honest with ourselves about our failures, and above all honest with the public about our failures, then we have shown that we can come back and we can win.

"We can win in 2020 if we are honest with ourselves and with the public and if we make the right decisions. If we take the right approach now, we will lay the foundations for our fightback and our next win."

Ms Cooper said she would shake up the "old boys' club" at Westminster if she won the race to replace Mr Miliband.

Labour has not had a full-time female leader, although Ms Harman and Dame Margaret Beckett have filled the role on an acting basis.

Ms Cooper told ITV's Lorraine "that's a glass ceiling that, of course, it's time for us to smash".

She said: "It's a chance to do politics differently, to challenge and shake up the Westminster old boys' club."

Ms Cooper's husband, former shadow chancellor Ed Balls - who ran against Mr Miliband in 2010 - lost his seat at the election but was finding solace in the success of his football club Norwich City after the team made it to the Championship play-off final.

Ms Cooper said: "It's hard and when you've worked for something and you've worked really hard for it and then it all stops ... it is tough but Ed is really strong and obviously Norwich getting into the play-offs has cheered him up a bit."

Labour veteran Frank Field said the party should stop taking money from the trade unions.

The former minister said: "I don't blame trade unions trying to influence the Labour Party but I think we should be very clear that they are one interest and one interest only."

He told BBC Radio 4's World at One: "I think we should move to a state where we have no trade union funding whatsoever. That, I think, itself would force the Government to face up - with them drawing large sums of money from donors - to the question about how do we fund political parties so that we are not tied to particular vested interests."

Mr Field said the next leader should have a "new conversation with the trade unions".

"Instead of them appearing to bully us, we should say we have got five things we want you to achieve in this coming parliament," he said.

Labour should challenge the unions on pushing for the living wage and raising productivity, he said.

"We give them a constructive agenda and get away from this agenda which makes them look as though they are bullying the Labour Party."