Sajid Javid has refused to defend Sir Geoffrey Cox amid new allegations that the former minister used his parliamentary office to carry out his legal work.
Labour has called on the standards commissioner to investigate whether the Torridge and West Devon MP breached the code of conduct, by using his office for work including advising the government of the British Virgin Islands in a corruption inquiry, saying it was an "egregious, brazen breach of the rules".
The Health Secretary declined to comment on "any individual case", but said it was important for MPs to be "completely open and transparent" about second jobs.
Asked if MPs should use their office for outside work, Mr Javid said "no", adding: "If someone believes for good reason that they might [have done], then I think what they should do is ask the appropriate authorities to look into it and they can get down to the facts.
"But of course you shouldn't be using, whether it's your parliamentary office... parliamentary stationery or anything that's funded or supported by the taxpayer," he added. "I think the rules are clear and of course all MPs would be expected to observe that at all times."
Chris Bryant, the Labour MP who chairs the Commons' standards committee, told BBC Breakfast: "It's a really important, I'd have thought, kind of basic rule."
Sir Geoffrey has been contacted for a comment.
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While some Labour MPs have accused Boris Johnson of returning to Cop26 early to avoid Westminster, Andy Burnham has said the "whole government operation" should be in Glasgow.
The Greater Manchester Mayor said: "I don't think our Government, or any government actually, has put policies on the table that will really connect with with people. We're in danger of missing that sort of big point of Cop; I would say I'm glad that the Prime Minister is coming here today - that's positive - but I would say this should be the whole Cabinet coming today.
"It should be a whole government operation, sending the message to the rest of the world that the UK is trying to lead this."
Mr Burnham added: "It doesn't look to me like we're really driving the leadership necessary here to get that big agreement that we need across the world and time's running out. I would say to the Prime Minister: get your Cabinet up here as well."
Sajid Javid has said he regrets having voted for the Owen Paterson amendment last week - going further than any minister so far.
Although colleagues including Dominic Raab and Nadhim Zahawi have said they regretted the "conflation" of the vote with wider standards reform, the Health Secretary expressed regret for having backed it last week.
He told Sky News: "Look, I voted for that motion, I do regret it, because I think it was wrong for me to conflate the Owen Paterson case with the right of an appeal.
"I do think it's important for parliamentarians on a cross-party basis to think about how they might always be able to improve a system, but it was wrong to conflate it with Owen Paterson.
"So I regret having voted for that. I'm glad it's something that is settled now as I believe yesterday the Government has brought forward a motion to rescind that and to accept the Owen Paterson report in full and I think that's the right thing."
Gordon Brown has said Parliament must address three key issues to avoid being overshadowed by allegations of sleaze.
The former prime minister told BBC Radio 4's Today programme said parliamentary offices should not be used to "advance a private cause", warning the current row was "bringing the House of Commons into disrepute... and people are losing trust in British politics"
The trio of problems include MPs' conflicts of interests, "dubious appointments" to the House of Lords and the "bigger issue" of foreign money entering the system.
"Unless the Prime Minister takes a grip of the issue, this Parliament will be remembered for the extent to which sleaze has allowed to become a feature of British politics again," he added.
Constituents will judge whether their MP "has done a good job or not" by how they vote, Sajid Javid has said.
The Health Secretary told BBC Radio 4's Today programme: "I can't speak to Geoffrey, but in my case, the external interests I had took about one to two days a month, so the vast majority of my time was spent on parliamentary affairs.
"As long as the MP has been completely open and transparent about it it is something for constituents to judge on - they can decide when it's time for an election.
"As long as the MP is transparent they can make a judgment about whether that MP has done a good job or not."
Sajid Javid has been forced to defend his own external interests, having worked for JP Morgan when he was a backbench MP.
Asked how that benefited his constituents, the Health Secretary said: "First of all, I was very open and transparent about it. you have picked one piece of external work i did and that is quite right, but I did others.. it is right that MPs, as long as they are following all the rules, can have outside interests."
Challenged again, he said: "It benefits... as a general thing, for MPs to be able to have external interests, it is good that it is not just all about politics."
Sajid Javid has conceded that MPs should not use their parliamentary offices to carry out second jobs, but refused to comment specifically on claims being made against Sir Geoffrey Cox.
The Health Secretary told BBC Radio 4's Today programme: "As a general rule, no one should be using things that are funded by the taxpayer - whether that is the parliamentary office or anything else - for personal gain.
"Someone might take a phone call or use their own phone for video call or whatever, but as a general rule that should not be happening."
He stressed that MPs should be "completely open and declare it in the proper way" if they did have outside interests.
The World Health Organisation's special envoy on Covid has criticised the "politicised" response to face masks, after Boris Johnson was pictured walking through a hospital without wearing one.
Dr David Nabarro told Sky News: "I'm not sitting on the fence on this one - where you've got large amounts of virus being transmitted, everybody should do everything to avoid either getting the virus or inadvertently passing it on.
He added: "We should do it all and we should not rely on any one intervention like vaccination on its own. So, please, would every leader be wearing face masks? Particularly when in indoor settings.
"This virus is unforgiving and we need to do everything possible to prevent it getting in between us and infecting us."
The "toughest thing about Covid" was how public health measures had become "politicised - in some places if you wear a mask, you're taught to belong to a particular political party", he added.
Sajid Javid has said compulsory jabs for NHS staff will not lead to "perfection ... we don't live in a risk-free world".
Challenged over the fact that people who have received the Covid vaccine can still transmit the virus, he came out all guns blazing, insisting that it reduced rates by between 65 and 80 per cent, depending on which jab people have had.
The Health Secretary added: "I'm not pretending, nobody is saying that you could have a 100 per cent method of protecting them, but what you can do is reduce the chances of them becoming infected.
"I think it is perfectly reasonable to say to everyone that works in health and care, please take this vaccine - some 93 per cent have had at least one shot and that's fantastic to see. There's still around 100,000 we estimate in the NHS that have not."
Sajid Javid has said he doesn't want "to see anyone lose their job, I don't want to see anyone have to walk away from their job" but defended the decision to impose mandatory Covid jabs on frontline workers.
The Health Secretary told Sky News: "It is important to be doing everything we can, because also most of all what I don't want to see - is someone ... that's vulnerable being exposed to Covid when it could have been prevented and perhaps they might even die from that.
"I think that would be totally unacceptable," he added.
"If you've been double-jabbed, then you're less likely to be infected."
Sajid Javid said "it's our duty" to make sure patients are as safe as they can be.
The Health Secretary told Sky News mandatory Covid vaccines are "about patient safety", adding: "Patients in hospital, in care, these are some of the most vulnerable people in our society and if they are exposed to Covid it could be fatal. I think it's our duty to make sure that they are as safe as they can be.
"That's why I announced this yesterday, so that from April next year, that if you work in the NHS or care, you will be required to be vaccinated and I want this to be a positive decision for everyone and to work with them.
"There will be absolutely no scapegoating, no singling out, no shaming, but working with colleagues to support them in every way that we can to help them make this positive decision so they can protect themselves, their colleagues and, most of all, their patients."
Flying is not the “ultimate evil”, the Transport Secretary has said, as Boris Johnson is set to return to Glasgow for crunch climate change talks.
Speaking about plans to reach net zero emissions and unveiling progress on proposals for greener transport, Grant Shapps said travel should be “guilt-free”.
His comments came as the Cop26 conference was expected late on Tuesday to publish a draft statement on what had been agreed.
However, some of the progress so far in Glasgow has been branded “disappointing” by commentators who said the US and EU were “failing to step up”.
On Tuesday night, the Prime Minister urged negotiators to “pull out all the stops”
Boris Johnson might have hoped that recess would give him some breathing space and cool down tensions among his backbenchers - but there is no sign of things calming down today.
There are fresh allegations against former minister Sir Geoffrey Cox, a vice chairman has resigned and questions being asked about whether a ministerial "fall guy" needs to go.
The Prime Minister is heading back to Cop26 today, but things aren't looking much rosier in Glasgow.
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