"Last week, the current home secretary admitted the Home Office sometimes loses sight of the individual".
And while last week's session was explosive, with the Prime Minister forced to play dirty just to get through it relatively unscathed, this was comparatively flat until Ms Cooper's intervention.
The revelations have prompted fresh criticism of the Government's "hostile environment" policy to tackle illegal immigration.
Amber Rudd said she felt "bitter regret" at her failure to grasp the full extent of the Windrush scandal as she revealed that a helpline set up by the Home Office had been overwhelmed with thousands of calls.
Ms Rudd will face questions from the Commons Home Affairs Select Committee later on Wednesday.
Asked if she had told May to ditch the target, Rudd said: "I have not discussed that with the prime minister". "Isn't it time she took responsibility and resigned?" Ms Rudd, who has faced calls to resign, told MPs that she became aware of the "potential issue" over the past three or four months as individual cases emerged.
"Although the Home Secretary has said that the Windrush scandal should cause a culture change in the Home Office, we still don't have clear answers about how things went so badly wrong or how something similar could be prevented in the future".
Asked on Wednesday whether the target led to people who have a legal right to live in the United Kingdom being targeted for deportation, Rudd said: "I don't think that's got anything to do with it".
And she denied that the target to bring down net migration to below 100,000 had fuelled the fiasco. "The problem here is that people were not properly documented", she said.
Corbyn also quizzed May on plans to compensate Windrush migrants who lost jobs, pensions or benefits after being wrongly targeted by immigration officials.
"I look back with hindsight and I'm surprised I did not see the shape of it sooner", Rudd said.
'I didn't see it as a systemic issue until very recently'.