Rishi Sunak told broadcasters on Thursday that ministers acknowledged the past week had not been good for the Government.
It comes amid signs of cracks between the Tory old guard and newer MPs – many of whom have much slimmer majorities than their longer-serving colleagues – over the perception of the saga by the public.
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Issues around standards in public life have come to a head following the resignation of former Tory MP Owen Paterson after he was found to have lobbied for two companies paying him for than £100,000 a year.
Last Wednesday, MPs were ordered to vote for a new committee to consider an altered system of appeals and to review Mr Paterson’s recommended suspension, only for ministers to backtrack hours later after opposition parties refused to co-operate.
Sir Geoffrey Cox, the MP for Torridge and West Devon, has since come under scrutiny for earning hundreds of thousands of pounds as a top QC while also an MP, including representing the British Virgin Islands in a corruption probe brought by the UK Government.
Mr Sunak told broadcasters on Thursday morning: “People will have different motivations for doing what they do. The pay is set by an independent body, that’s absolutely right.
“And with regard to second jobs, there’s an independent process that we have that’s set by Parliament that governs all of those things. And it’s absolutely right that that process is followed to the letter.”
But he added: “Now look, on the broader point – and just reflecting over recent events – I think for us as a Government, it’s fair to say that we need to do better than we did last week, and we know that.”
Writing for Times Red Box on Thursday, Peterborough MP Paul Bristow, who was elected at the last general election, said “many Conservative colleagues” had shared his reservations over the Paterson affair.
Mr Bristow, who has a majority of 2,580 compared to Sir Geoffrey’s 24,992, defied the party whip to vote against the Government last week and added: “We are now faced with the task of restoring public confidence.”
Business minister Paul Scully was asked whether MPs can deliver for their constituents while working in the British Virgin Islands (BVI), as Sir Geoffrey did.
He told LBC: “I’m not going to speak on behalf of what Geoffrey Cox did there or otherwise.
“You clearly don’t need to be in the constituency day in, day out, but nonetheless you need to be available to your constituents and contactable by your constituents.”
Pressed on whether the Chief Whip authorised Sir Geoffrey to work from the Caribbean while voting by proxy in the Commons, Mr Scully replied: “I don’t know the ins and outs of it, I don’t know the whipping situation.”
A spokesman for the Chief Whip Mark Spencer confirmed on Wednesday that Sir Geoffrey had been granted a proxy vote, but not whether it was known Sir Geoffrey would be in the BVI.
Speaking on Sky News, Mr Scully said: “I’m not going to defend Geoffrey or say anything – that’s up to Geoffrey, it is between him and his voters.”
Asked whether he could see that the “optics were not good” in relation to the second jobs debate, Mr Scully replied: “Absolutely, I can see how it looks.
“It is really regrettable that we’ve got to this situation.”
Pressed on what was regrettable, he added: “That we’ve got in a position that, first of all, last week we allowed the situation for one individual [Owen Paterson] to be conflated with the standards procedures, which do need looking at, they do need an appeals process, but in order to do that we need a cross-party consensus.
“Then, as I say, for a number of other issues to be conflated which we need to tackle, but we need to tackle in a level-headed way and certainly not to get into a party political bun fight.”
Sir Geoffrey has defended his outside work and said it was up to his constituents to judge whether they were happy with the situation.
It comes as Lord Evans, chairman of the Committee on Standards in Public Life, said that if an MP is spending a “huge amount of time” on a second job, they would not be able to fulfil their duties to constituents.
Speaking to BBC Radio 4’s Today programme about his committee’s 2018 recommendations on MPs’ external work, the peer said: “We said that the critical thing was that nothing that an MP does should get in the way of their ability to work in support of their constituents – so the amount of work they do, the sort of work, needs to be judged against that.
“If somebody is spending a huge amount of their time on a second job, then they can’t be maintaining support for their constituents.”
Asked whether it matters more about how much time an MP spends on their outside work rather than how much they are earning, Lord Evans replied: “I don’t think it is ultimately a matter of how much, I think it is a matter of is it clear that the MP who has been elected by their constituents, that their main focus, their main priority is on being the best MP that they can be?
“And that takes time and that takes concentration.”