But she admitted her department still had no idea how many migrants have been wrongly detained by immigration authorities - like the grandmother Paulette Wilson, who spent a week in detention after being told she was in Britain illegally.
The Labour leader failed to convert an open goal last week on the Windrush scandal, which runs to the heart of Conservative immigration strategy and straight to Mrs May's record as Home Secretary, and a week more of heartbreaking case studies and damaging headlines gifted him another chance.
"At last she's been forced to act upon it", Corbyn said.
She also denied that individuals were targeted for deportation.
The crisis has focused attention on the role of Prime Minister Theresa May, who as interior minister set out to create a "really hostile environment" for illegal immigrants, imposing tough new requirements in 2012 for people to prove their legal status. The problem here is that people were not properly documented'.
Mrs May did not even have to attack Mr Corbyn over anti-Semitism in his party, and somehow emerged for her final answer laughing and smiling, after the Labour leader chose suddenly to switch targets to current Home Secretary Amber Rudd.
And then to the delight of previously moribund Labour backbenchers: "A few years ago the Prime Minister said "I'm actually sick and exhausted of a government minister who simply blames other people when something goes wrong".
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They face hosts England at Cardiff, where the Tigers had defeated Australia and New Zealand. The Olympic Stadium looks set to miss out on hosting matches at next year's World Cup.
"Had the Windrush generation not mounted the campaign that they have, had members on this side of the house not raised the matter persistently, there would be no compensation, there would no be no review, there would be no apology", he said.
She added: "I have private conversations with the prime minister which will stay private".
Ms Cooper said the Prime Minister had been "hiding" behind ministers and officials in the wake of the damaging debacle, which saw British residents mistakenly threatened with deportation.
"Do not try to hide behind the cabinet when they don't agree with you on this and are trying to clear up the mess".
But she said: "I bitterly, deeply regret that I didn't see it as more than individual cases that had gone wrong that needed addressing".
Speaking earlier on Wednesday, the children and families minister, Nadhim Zahawi, pledged that all Windrush cases would be dealt with within two weeks. "I think that has to be the focus", he told BBC Radio 4's Today programme.