Mrs Hodge says she's passionate about the job
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Margaret Hodge says she did consider stepping down as children's minister after she falsely branded a sex abuse victim "extremely disturbed".
Mrs Hodge said she had "made a big mistake" in the remarks she made about victim Demetrious Panton.
But she told BBC's Question Time she decided to stay because "everybody in the children's world" had "expressed confidence" in her.
She has already made public and private apologies and a donation to charity.
Resignation calls
The minister made her remarks about Mr Panton in a letter to the BBC which was then broadcast by the corporation.
Mrs Hodge was complaining about an inquiry into a child abuse scandal in London's Borough of Islington - the council she used to lead.
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I made a bad mistake and I did apologise and in 30 years of public service ... you do make mistakes
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After Mr Panton threatened to sue, a deal was hammered out by his solicitors and those representing the minister in which she also agreed to write to him to explain her comments.
She has already agreed to give £10,000 to a charity nominated by Mr Panton and pay his legal costs.
During Question Time Mrs Hodge spent a number of minutes defending her position as two panellists and one audience member argued that she should have resigned.
She also told how she considered quitting her job.
Passionate believer
"I made a bad mistake and I did apologise and in 30 years of public service ... I hope all of us who have been in public service ... you do make mistakes," she said.
"Of course when I made this mistake and when I apologised, both in public and in private, I reflected on whether I could continue with that absolutely crucial job we have got of transforming services for children and in the end, the decision I came to - I do feel absolutely passionately about these issues.
"I really, really came into politics to change the world a bit and I really want to change the world and create better outcomes for children.
"My record as minister - and I've had responsibility for children's issues before - is good."
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I think Margaret is a perfectly good minister, but I think she has no credibility as children's minister
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Mrs Hodge ran through the children's policies she had implemented and stressed: "Of all the people I deal with in the children's world ... they have all expressed confidence in me.
"All I would ask everybody is to consider whether or not they in their lives have made mistakes, whether or not they have felt that inhibited them from going on."
But Tory chairman Liam Fox said while he thought Mrs Hodge was "a perfectly good minister", she had "no credibility as children's minister as a result of what happened".
"I think that to use the phraseology that she did and then be forced to apologise in court and then continue in that particular job leaves her weakened in a job that requires a strong minister," he said.
'Brass neck'
Mr Fox said there was a problem with ministers refusing to resign and a culture of "bullying" within government.
"It seems to be that if you get in the way of the current government, they will use bullying as the first trick to shut you up - I think that is very, very worrying."
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Personally I don't think she should resign
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A female audience member added: "When you accepted a Cabinet post, you also accepted ministerial responsibility - you have no credibility as a minister and you should therefore resign."
Writer and broadcaster William Shawcross, whose father, the late Lord Hartley William Shawcross, led the British prosecution at the Nuremberg trials, said: "I'm rather astonished by the brass neck of politicians."
Turning to Mrs Hodge, he said: "I think you are also a little bit too pleased with yourself, saying you have massive support from everyone involved in the children's world."
But Mark Oaten, the Lib Dems home affairs spokesman, said he disliked the immediate call for resignations when something went wrong.
Brave
"I think the minister made her position clear - she was foolish about the way she tried to manage the story with the BBC, but she's said sorry ... and I hope Margaret has learned from what has happened.
"Personally I don't think she should resign."
Shami Chakrabarti, director of human rights group Liberty, said Mrs Hodge had been "very brave" to come on the programme.
"I couldn't be director of Liberty if I believed in lynch mobs ... I would like to see her get another chance," she added.