Boris Johnson was last week cleared of misconduct and of breaking the ministerial code by his adviser on ministerial interests.
But Lord Geidt said he had "acted unwisely".
Labour says he might have broken the rules governing the conduct of MPs after failing to declare a donation.
Labour is keen to increase the pressure on Mr Johnson over the renovations to his flat as part of its wider charge of cronyism against the government, and has written to the Parliamentary Standards Commissioner, Kathryn Stone.
Last Friday, Lord Geidt - who was appointed by the PM in April as his ministerial standards adviser - said Tory donor Lord Brownlow had paid an invoice to cover some of the costs for the works, which were overseen by Mr Johnson's wife Carrie.
He accepted that Mr Johnson had not been aware how the bill had been paid but rebuked him for "unwisely" allowing the refurbishment to proceed "without more rigorous regard for how this would be funded".
At the time, Labour said it was "staggering" that Mr Johnson could "rack up" a bill "yet have no knowledge of how it was eventually paid".
Now, the party says Mr Johnson still has more questions to answer in his capacity as an MP, which falls under the remit of Ms Stone.
In the letter to the commissioner, Labour's Deputy leader, Angela Rayner, noted Ms Stone had previously reprimanded Mr Johnson in 2019 for failing to register a share of a property within the required time frame, and warned any further breaches may warrant "more serious sanction".
But Ms Rayner said "far from learning the lessons of his previous transgressions, the prime minister has continued with his attitude of treating basic standards of integrity, openness and transparency with contempt, and behaving as though there is one rule for him and another for everyone else,".
She says the fact that Mr Johnson told Lord Geidt he became aware of the donations for the works on the flat in February this year but did not settle the invoices personally until 8 March suggests he is in breach of parliamentary rules that all MPs must follow.
The Code of Conduct for Members of Parliament states that all MPs must register financial interests, donations and gifts, and Mr Johnson, as prime minister, is not exempt from that.
Ms Rayner added: "I therefore write to request that you instigate a further investigation into the prime minister's failure to register this donation, using your powers as commissioner."
She also accused Mr Johnson of breaching the general principles on conduct in public life, including "selflessness", "integrity" and "leadership".
Ms Stone is already investigating whether Mr Johnson properly declared a £15,000 holiday on the Caribbean island of Mustique with his wife Carrie.
A Downing Street spokeswoman said: "Lord Geidt's independent report shows the prime minister acted in accordance with the ministerial code at all times and sets out the background to the intended establishment of a Downing Street trust.
"The report shows how, at all times, the prime minister followed the advice of officials and he has made a declaration in his list of ministerial interests, as advised by Lord Geidt.
"The House of Commons rules and Electoral Commission guidance are clear that support relating to ministerial activity should be declared through ministerial declarations and this has been done."
The Parliamentary Commissioner for Standards has the power to refer a case to the Committee on Standards if she deems it is serious enough and the committee can, in principle, recommend suspension.
The sanction for less serious cases can require the MP to acknowledge and apologise for any breach.
Mr Johnson faces two other inquiries in to the flat refurbishment.
The Electoral Commission is conducting its own investigation in to whether the Conservative Party broke the rules on declaring donations over the Downing Street flat and has the power to issue a fine of up to £20,000.
Meanwhile, Cabinet Secretary Simon Case, the UK's top civil servant, is also looking into the refurbishment, including whether donations were properly declared.