A completely independent investigator on ministerial standards is needed, Labour has said, as the row continues over the refurbishment of Boris Johnson’s private living quarters in Downing Street.

Liz Kendall, the shadow care minister, said that the current arrangement, under which the adviser on ministerial standards has no power to commission his own inquiries, gives the prime minister too much power to decide on how investigations are determined.

“We need an absolutely independent adviser on ministerial interests who has the power to initiate investigations,” Kendall told BBC Radio 4’s Today programme.

“Let’s not forget that the last one quit when the prime minister overruled his finding that Priti Patel had bullied staff. It’s been months since the new one has been appointed, and saying that Lord Geidt can only suggest investigations take place isn’t good enough. It’s not just that Johnson would be marking his own homework, he wouldn’t be setting his own homework in the first place. We need to have complete openness, honesty and transparency if we [are going] to have confidence in the system.”

Christopher Geidt, the former private secretary to the Queen who has been appointed as Johnson’s new adviser on ministerial standards, is to immediately look into the controversy over payments for the renovation of Johnson’s Downing Street flat.

But there is concern that he is not sufficiently empowered to hold the prime minister to account if he finds there has been any wrongdoing, after Johnson’s official spokesperson said the PM would remain the “ultimate arbiter” of whether the ministerial code has been broken, even if the investigation centred on himself.

Prime ministers get a budget of up to £30,000 a year to renovate their Downing Street residency, but newspaper reports have suggested Johnson has spent up to £200,000 on a refurb, to be overseen by the interior designer Lulu Lytle.

Last week, the Daily Mail published details of an email from the Tory peer Lord Brownlow, in which he said he was making a £58,000 donation to the party “to cover the payments the party has already made on behalf of the soon-to-be-formed ‘Downing Street Trust’”.

Geidt’s investigation is one of three into how the refurbishment of Johnson’s flat has been paid for, with the cabinet secretary and the Electoral Commission also carrying out inquiries. Johnson last week said he had paid for the refurbishment himself. He has insisted he has not broken any laws over renovations.

The vaccines minister, Nadhim Zahawi, spent Thursday morning visiting broadcast studios to defend his boss’s actions and Geidt’s role. He said the prime minister was prepared to make any further declarations about the refurbishment required by the new ministerial standards adviser.

“The prime minister has been very clear that he paid for this, that he also followed all the ministerial code, he took advice and if Lord Geidt, in his investigation, requires the prime minister to make any other declarations, then he will also do that,” Zahawi told Times Radio.

Zahawi avoided answering the question, put to him on Today, of whether Johnson was always going to pay for his new cushions and wallpaper. He said: “I think I’ve answered plenty of questions on this. The prime minister paid for the bill, the prime minister has followed the ministerial code, the prime minister has appointed Lord Geidt and has asked Lord Geidt to look into this and if there’s other declarations that need to be made, he will make them.”