Ken Clarke has suggested that David Cameron did “some sort of deal” to win the support of Rupert Murdoch’s newspapers in the run-up to the 2010 election. According to Clarke, in his evidence to the Competition and Markets Authority, when – at Cameron’s instigation – he held a meeting as justice secretary with Rebekah Brooks, she “described herself as running the government now in partnership with David Cameron”.

Clarke says that she tried to use this influence to get him to introduce prison ships. She failed in that particular lobbying effort – which turned out to be good news for some of her subsequently imprisoned former colleagues. But the Murdochs were interested in more than just prisons policy – they always have been.

Murdoch got his man – former News of the World editor Andy Coulson – into the heart of the Conservative operation, first as the Conservative party’s head of communications, and later at Downing Street. “That was part of the deal I assume,” says Clarke. Cameron was prepared to believe Coulson’s denials that he had had any involvement in phone-hacking – a credulity that he later came to regret.

When it all came crashing down in 2011, with the phone-hacking scandal, the closure of the News of the World, and the – temporary, as it turned out – abandonment of the Murdochs’ bid to take over Sky, it looked as if the cosy deal between the Murdochs and the Conservative party might have to end. We learned about a corporate culture within the Murdoch empire – of documents deleted, of ethical standards conveniently suspended, of blind eyes turned to criminality – that nobody could defend. Suddenly, association with Rupert Murdoch was a political liability.

I’ve watched with fascination over the last few years as the Murdochs have tried to claw themselves back into the privileged role they so deservedly lost. Their papers backed the Tories (and, interestingly, the SNP) in the 2015 election, with some success. But this year has shown their powers fading. They threw everything they could at Labour, and at Jeremy Corbyn, in the general election – and it didn’t work.

On election day in June, the Sun’s front page screamed: “Don’t chuck Britain in the Cor-bin”. Millions of voters ignored it – and deprived Theresa May of the majority she’d taken for granted. For once, and perhaps, for ever from now on, it wasn’t the Sun wot won it.

Since then I’ve praised Karen Bradley, the culture secretary, for doing the right thing and referring 21st Century Fox’s bid to take over Sky to the Competition and Markets Authority – a decision I don’t believe the Murdochs expected for one moment. Maybe she – like Corbyn – feels liberated by the election result. If the Murdoch papers can’t deliver on their side of the bargain for the Tories, why should she bend over backwards to do what they want, instead of following due process?

But Theresa May still faces questions about her relationship with Rupert Murdoch. We still don’t know what was discussed at her private meeting with him in New York in September 2016. And when I wrote to her in June, following a tip-off, to ask her if it was true that Murdoch had asked her to reappoint Michael Gove to the cabinet, I was surprised to receive a reply that refused to deny the allegation. It would, after all, have been so easy to deny it.

The Murdochs are still seeking to expand their empire, even as new revelations about corporate behaviour pile up. Their flagship TV station, Fox News, has been hit by a rolling sexual harassment scandal – in which we now know allegations were covered up, accusers paid off and alleged perpetrators handed new multimillion-dollar contracts, despite the company’s knowledge of serious sexual misconduct claims. I met some former Fox News employees in parliament this week and was shocked at the stories they told me.

In just the last few months here in the UK, Murdoch companies have paid out damages to a former army intelligence officer whose computer was hacked by private detectives working for the News of the World, and settled 17 cases of phone hacking and illegally obtaining personal information. And more cases are outstanding. The very fact that this legal process is lumbering on shows that the Murdoch empire is still unwilling to face up to what happened inside its newspapers: every new revelation has to be dragged from them.

The scandal hasn’t ended, as Clarke’s testimony shows. That’s why I still believe we need to see part two of the Leveson inquiry – promised in 2011 by all political parties – to look into the extent of unlawful and improper conduct within News Group and other media organisations. I hope the Conservative party can show that it’s no longer in the business of doing deals with the Murdochs, and get on with it.

• Tom Watson MP is deputy leader of the Labour party