It has been a mixed week for the Labour leader. Having reversed his counterproductive decision to refuse to meet the prime minister to discuss Brexit, he took the initiative with a proposal for a closer economic relationship with the European Union after Britain leaves – a plan that was welcomed by Donald Tusk and that put Theresa May under pressure from the soft-Brexit wing of her own party.

Naturally, Jeremy Corbyn’s demand for a permanent customs union did not go down well with The Independent, or with Labour MPs who support our campaign for a Final Say referendum, because it seemed to confirm that he wants to leave the EU without going back to the people – hence his back-pedalling in a speech in Coventry today, saying Labour could still back a new referendum.

Mr Corbyn continues to try to keep both sides of his party together, while pleasing neither. He had more success, however, in dealing with the problem of antisemitism in the party. The issue resurfaced on Monday when Labour MPs unanimously agreed a resolution at their weekly meeting calling on the leadership to step up its efforts to deal with the problem.

It was more than unfortunate timing, therefore, when it later emerged that Wavertree Labour Party was planning to convene a special meeting to debate two motions of no confidence in its MP, Luciana Berger. Formally, there is no connection between the local party’s political differences with Ms Berger and the antisemitic abuse she has received. The party supports Mr Corbyn and she does not.

But one of the reasons she disagrees with the Labour leader is that she believes he is not serious about fighting antisemitism, so it is not possible to take the fact that she is Jewish out of the political disagreement with her local party. Especially when one of the members proposing a motion of no confidence has previously called her a “disruptive Zionist”.

We cannot be sure what happened next, but on Friday the no-confidence motions were withdrawn and the Wavertree Labour Party cancelled the meeting. Someone speaking for Mr Corbyn, but asking to be identified only as a source in the leader’s office, said: “It was the right decision.” It would be reasonable to assume that Mr Corbyn had intervened to persuade the local party that now was not a good time to be starting to deselect a Jewish MP.

If so, this is encouraging, although it would have been a little more courageous if Mr Corbyn could have said something on the record, instead of relying on press officers making deniable anonymous statements.

It was also encouraging that John McDonnell, the shadow chancellor, reacted so emphatically to a caller on a radio phone-in on Tuesday who described claims of antisemitism in the Labour Party as “smears”. Mr McDonnell told him: “It isn’t a smear campaign, I’ve seen the evidence, I’ve seen the stuff on social media.”

So that was two steps forward. It was a shame, therefore, that Mr McDonnell should have taken a step back three days later, when he appeared to blame Ms Berger for the conflict with her local party, suggesting that it was because she “has been associated in the media with a breakaway party”. After the way she has been treated by antisemites who claim to be acting in Mr Corbyn’s name, who could blame her for wanting to leave the party? But Mr McDonnell seemed to think the solution was for her to make some kind of declaration of loyalty.

If we could have more of the Mr McDonnell of Tuesday, and less of the Mr McDonnell of Friday; and if we could have more of the Mr Corbyn who intervened to persuade the local party to lay off Ms Berger, and less of the Mr Corbyn who hides behind press officers, all the better.