Jeremy Corbyn has accused the government of being “too weak and too divided” to properly protect the country during Brexit, as he used prime minister’s questions to lambast Theresa May over the issue before she heads to Brussels for a crunch summit.

The Labour leader cited warnings by businesses that a lack of clarity and the risk of border friction could hit investment and jobs, accused May of being obsessed by internal Conservative wrangles and urged her to keep the UK in the customs union.

“The Conservative party has spent two years arguing with itself instead of negotiating a deal in the public interest,” Corbyn told a sometimes raucous Commons. “And now, just days before the deadline, they’re still bickering amongst themselves.

“The prime minister and her government are too weak and too divided to protect people’s jobs, our economy or ensure there is no hard border in Northern Ireland. So the prime minister has a choice – she can continue to put the Tory party’s interests first or she can listen to unions, businesses, and put the interests of the people of Britain first. Which is it to be?”

May, who will address EU leaders at the European Council summit before dinner on Wednesday, responded by insisting she would deliver a good deal, and condemned Labour for holding open the possibility of a second referendum.

“Labour can play politics, the Conservatives deliver to the people of this country,” May said, after citing a list of claimed achievements on the economy and jobs, to cheers from her own backbenchers.

Corbyn used all of his allotted questions to badger May on her Brexit plans, beginning with a jibe at her recent habit of not referring to her Chequers proposal by name – seen as a way of trying to detoxify it for rebellious Tory MPs.

“Given the prime minister did not once mention Chequers either in her conference speech or in her statement to parliament on Monday, does this mean the Chequers plan is now dead?” Corbyn said.

May responded: “He asked me if the Chequers plan is dead. The answer is no.”

Asking also about reports that the UK could be obliged to pay the bulk of its so-called divorce bill to the EU even in the event of a no-deal departure, Corbyn then challenged May to spell out what she had agreed on the contentious issue of the backstop in the interim agreement with the EU.

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“In December the prime minister signed an agreement with the EU which stated: ‘In the absence of agreed solutions, the United Kingdom will maintain full alignment with those rules of the internal market and customs union.’ Will she confirm that this agreement still stands and that she signed up to it without any time limit?” the Labour leader asked.

May pointedly did not confirm this – she faces significant internal unrest, particularly over the time-unlimited nature of the backstop, something she is seeking to change. Instead, the prime minister said the main objective was a deal on the future relationship with the EU, which would mean the backstop was not needed.

She said: “We were clear then and we are clear now: the purpose of the backstop is to bridge the gap between the end of the implementation period and ensuring that the future relationship is in place.”

Leading to his final questions on jobs and the economy, Corbyn noted warnings on future investment and friction-free trade from the likes of Vauxhall and AstraZeneca.

He said: “Jobs are at risk. Why won’t the prime minister back a customs union, supported not only by Labour and trade unions but by businesses and by, I suspect, a majority in this house, to protect those jobs?”