Another day, another scandal on the Labour left. Today the person responsible for rooting out racism as chair of Labour’s internal disputes panel has resigned after her disgraceful support for a council candidate spreading Holocaust denial.

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But some on the left genuinely think all these accusations against them of antisemitism are just the latest weapons used by “Blairites” against Jeremy Corbyn and his circle. They are blinded to seeing how legitimate rage against the Israeli government and support for Palestinians leaches into the same anti-Jewish racist passions of, say, Hamas.

Trying to rescue Labour’s credibility from Corbyn’s too-little-too-late apology had Labour MPs out demonstrating with Jewish groups – and now they get threats of deselection. The Labour leader in Barnet expresses alarm at the damage done, fearing loss of north London Jewish votes risks Labour’s chance to seize this exceptionally rightwing Tory council in May. John McDonnell on Thursday morning tried hard to cauterise the wound. “We will deal with it firmly and severely,” he told the BBC’s Today programme. “We will eradicate it from our party.”

Let’s assume that happens and that any future outbreak is stamped on so harshly by Labour’s new general secretary that it never happens again. But for now the great political cleft in Labour just yawned wider. What’s to be done?

When Corbyn outshone expectations at the general election, the leadership question was killed stone dead. He will lead in 2022: under the Fixed-Term Parliament Act, this government can lose any number of votes and still not fall before then. There is no knowing who the next Tory leader will be – the fruitcake, the narcissist or some plausibly sensible face? Will we be out of the EU or still in transition – and if so to where? Wise commentators have thrown away their crystal balls.

The one certainty is that Corbyn will be leader, with unshakeable membership support. He will again take on the worst Tory government in living memory, whose Euro-madness has inflicted irrecoverable damage on future generations, whose austerity has impoverished millions, stolen life-chances from children and wrecked public services. Those worrying about Corbyn’s ability to govern, the weakness of his team, the lack of gritty detail backing up radical policies should never forget the appalling prospect of a newly invigorated Tory government in power until 2027.

The only course for every Labour MP is all-out assault on this government. And since the election, so most have done. Labour’s chairs of select committees have been exceptionally good at holding May to account: Yvette Cooper at home affairs, Rachel Reeves at business, Hilary Benn at Brexit, Frank Field at work and pensions, Meg Hillier at public accounts. These non-Corbynites can’t be accused of slacking in exposing government failure. They are daily reminders of Labour’s depth of brain, energy and experience.

Now Corbyn has control of every powerbase within the party, he and his coterie should reject the ceaseless search for enemies within, from those such as Skwawkbox, and open up his small group of the faithful. Widening out to embrace the breadth of his own party would bring in the voters Labour needs. Are they alarmed enough that Labour still only level pegs with this exceptionally inept government?

As for the other side, extreme outriders such as John Woodcock do real harm, along with a handful of Labour people who might prefer the party to lose than Corbyn to win. Owen Smith was right on substance – on the single market and an option for a public vote on the deal – but his stand was mistimed. Why now, as Labour inches forward in the right direction?

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McDonnell again today reinforced Labour’s Brexit line, after Emily Thornberry’s misinterpreted throw-away remark: Labour sticks to Keir Starmer’s six strict tests. Since none of them can be met by May’s spilt party, not the Irish border, nor “the exact same benefits” of customs union and single market, Labour will vote against the government on every Brexit issue. Never has an opposition been in such a strong position to rescue the country from a government’s disastrous errors.

For both sides of the Labour party there is no alternative but cohesion. Can they do it? If Corbyn keeps leading from inside a tiny group of the like-minded, Labour will fail to win the wider votes it needs. All distractions from that goal are a dereliction of duty. The worst accusation against either side would be that they preferred fighting internal wars to confronting the real threat to the country on the opposite side of the house.

• Polly Toynbee is a Guardian columnist