Just as his path to The Lodge was clearing, Peter Dutton’s chances of usurping Malcolm Turnbull were dealt a blow, with claims he may have fallen foul of the constitution.

Ten News reported on Monday night Mr Dutton has an interest in two Brisbane child care centres through a family trust.

From July 2 this year child care centres receive a direct subsidy from the federal government.

This could put him in breach of section 44 of the constitution, which says anyone who has “any direct or indirect pecuniary interest with the public service of the commonwealth” is ineligible to sit in Parliament.

Two leading constitutional lawyers, Anne Twomey and George Williams, said it was possible Mr Dutton had infringed section 44.

The minister’s office denied he has a case to answer.

“Mr Dutton’s legal advice clearly states there is no breach of section 44,” a spokesperson told numerous media outlets.

The immediate suspicion was that Turnbull supporters had leaked the story to head off a leadership challenge by Mr Dutton as early as Tuesday’s scheduled party room meeting. This was heightened when Labor denied any responsibility for the story.

But the journalists who broke the story, the highly respected Hugh Riminton and researcher Kate Doak, insisted there were no leaks, only their own careful investigation of Mr Dutton’s business interests and disclosures to Parliament.

Deputy Prime Minister Michael McCormack admitted Mr Dutton might have to face the High Court to clear his name.

“If there is enough evidence and the matter needs to be referred to the High Court, well, I’m sure, as the usual process is, it will,” Mr McCormack told ABC radio on Monday night.

Mr Dutton railed against Labor’s weak border security policies during question time. The constitutional claims may rattle his confidence. Photo: AAP

When asked if Mr Turnbull would still be the prime minister by the end of the week, the Nationals leader replied: “Oh of course he will. There you go. There’s a really simple answer to the question.”

Apart from this glimmer of hope, Monday was a miserable day for the Prime Minister, as he squared off against the media and Labor to explain his decision to jettison an emissions target – the only part of the fraught National Energy Guarantee that targeted climate change.

His chosen line: “The policy … has been improved following consultation.”

And what about global warming? “We are taking real action on climate change,” Mr Turnbull told Parliament. “We are.”

But any mandatory emissions targets had been, in the words of the Deputy Prime Minister, “shelved”.

Numerous experts slammed the dumping of the emissions target, warning it would push up power prices and create more instability in the energy generation system.

Labor’s attack was to claim the government had made no real attempts to overcome recalcitrant conservative backbenchers by securing bipartisan support for the NEG policy.

The shadow cabinet also pointed out, gleefully, that the Turnbull government had now cycled through five different energy policies: an emissions scheme (dumped in 2016), a clean energy target (dumped in 2017), and three versions of the NEG.

There may be more policy ‘improvements’ to come. Still hanging over Mr Turnbull’s head is what to do about the seemingly unpopular big-business tax cuts, which were blamed for the Coalition’s resounding defeat in the Longman by-election in Queensland. It was this drubbing that kicked off the leadership speculation.

The tax cuts for businesses with annual turnover above $50 million (including the much-maligned banks) came before the Senate on Monday night, but a final vote was deferred. Not that the result is in doubt – the government has failed to secure the necessary crossbench votes to overcome the opposition of Labor and the Greens.

There is rampant speculation that the policy – yet another signature pledge of Mr Turnbull – will also be jettisoned later this week, to help the Prime Minister keep power and fight Labor.