Senior ministers have rallied round beleaguered Amber Rudd after she admitted she should have known about official targets for removing illegal immigrants.

The home secretary did not see the internal Home Office memo that boasted of exceeding the department’s target for immigration removals, Michael Gove has claimed, as he suggested that Labour had “weaponised” the Windrush scandal to distract from its own row over antisemitism.

The environment secretary, speaking on BBC Radio 4’s Today programme on Saturday, said Rudd did not receive the document despite it having been copied to her and Brandon Lewis, then immigration minister, and several civil servants and political aides.

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The six-page memo, revealed on Friday by the Guardian, says the department had set “a target of achieving 12,800 enforced returns in 2017-18”, boasts that “we have exceeded our target of assisted returns”, and adds that progress had been made on increasing enforced removals by 10%.

The issue has become particularly toxic because of coverage of the Windrush generation of migrants from the Caribbean. Many have been left destitute or homeless, and denied benefits and healthcare because of the Home Office’s “hostile environment” policies targeting migrants.

Backing Rudd’s own claims on Twitter that she had simply missed the document, Gove said: “What we are witnessing is an example of government not functioning as it should have done, and that’s something for which the home secretary has taken responsibility.

“There does seem to be a series of leaks, or sharing with the Guardian in particular, that is designed to serve a particular agenda … There’s a campaign against the government and against the home secretary. What’s not surprising is that this happens at the same time as the Labour party is mired in allegations of it’s failure to deal with antisemitism.”

“This is about politics,” Gove said. “And the focus on whether or not a particular document that was cc’d to a particular address was then put in a particular box at a particular time – and we know it wasn’t – is intended to distract from the difficulties that the Labour party faces with handling prejudice in its own ranks.”

“Labour are attempting to weaponise this. I think that is quite wrong.”

The justice secretary, David Gauke, said Rudd was an “excellent” home secretary who had accepted she made a mistake.

“She didn’t knowingly mislead the House of Commons, but she accepted that she was inaccurate in her statements,” he told the BBC Radio 4 Today programme.

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But the shadow home secretary, Diane Abbott, said Rudd should have been prepared to accept responsibility for what had happened. “I am just surprised that she doesn’t seem to take the issue seriously enough to offer her resignation,” she told Today.

The decision to set a “broad numerical target” could have contributed to the Windrush fiasco, Abbott said. “It wasn’t saying, for instance, we have to have a target for deporting former criminals. The danger is that that very broad target put pressure on Home Office officials to bundle Jamaican grandmothers into detention centres.”

Rudd is to be recalled to give evidence to the Commons home affairs committee, after she responded to the leaked document in a series of tweets late on Friday night insisting she had not seen the leaked memo, “although it was copied to my office, as many documents are”.

She repeated her claim that she “wasn’t aware of specific removal targets” and said: “I accept I should have been and I’m sorry that I wasn’t.”

The Labour chair of the home affairs committee, Yvette Cooper, told Today: “We have obviously been given inaccurate information to parliament twice now. This is a serious concern and I am calling Amber Rudd to come back and give further evidence to the committee.”

The Labour leader, Jeremy Corbyn, who was one of only 18 MPs to vote against the 2014 Immigration Act at the root of the scandal, said Rudd had misled parliament.

“There has been target figures put up in offices all over the country of what staff should achieve, and she’s telling us that there were no targets,” he told Sky News on Saturday. “This is more than odd. And so when she makes her statement on Monday I hope she’s going to say that because of all this she is going to offer her resignation.”

Rudd and the prime minister, Theresa May, who was home secretary at the time of the act, had been told the new law was dangerous and would put the Windrush generation in a precarious situation, Corbyn said.

“That was ignored, and now the government is having to offer compensation to the Windrush generation. But they should think for a moment of the stress that often older people have gone through, who have worked their whole lives in this country, consider themselves to be full British citizens – which is exactly what they are – and they should be treated with the [same] dignity and respected as every other citizen should be treated.”