The culture minister, Maria Miller, tried to limit an investigation into her expenses by warning the parliamentary standards commissioner, Kathryn Hudson, that she was acting outside the law and threatening to refer her to a Commons committee.

In a sign of Miller's attempts to prevent an investigation into her claims for interest payments on her mortgage, she told Hudson that the commissioner would be acting in a way that was "unwarranted, unfair and contrary to all standards of due process and legality".

Miller, who was ordered on Thursday to apologise for her conduct towards Hudson and to repay £5,800, even threatened to refer the watchdog to the "supervisory jurisdiction" of the House of Commons standards committee.

The lengthy and fruitless attempt to limit the investigation emerged in the correspondence between Miller and Hudson over the MP's expenses.

On Thursday Miller was told by the Commons' standards committee, which issues judgments on the watchdog's report, that she would only need to repay £5,800 – a reduction from the £45,000 Hudson recommended that the minister return for over-claimed allowances on her second home in Wimbledon, south-west London.

The Daily Telegraph reported in December 2012 that Miller claimed more than £90,000 in mortgage interest and other costs for the property in which her parents lived even though parliamentary rules said a second home must be for the exclusive use of MPs. The property was bought for £234,000 in 1996, but she and her husband subsequently raised a mortgage on it of £525,000.

The decision by the standards committee to decrease the amount suggested by Hudson prompted the Labour MP John Mann, whose complaint caused the inquiry into Miller, to say on Friday night that politicians should no longer act in judgment over themselves.

He told the Guardian: "Self-regulation of MPs by MPs is now well and truly dead. The committee should only exist to look at how parliament can improve standards in public life. Instead the independent commissioner should make decisions on MPs with appeals to the independent advisers. And the electorate needs to be give the right of recall for serious offences."

The row over Miller's expenses intensified on Friday as:

• It emerged that the committee was planning to ask Miller to repay considerably less than the £5,800 finally agreed, but settled on that figure when Miller proposed it.

A source told the Guardian: "It was thought that she owed less, but the committee went along with her calculation. It suited them. They thought it was a good deal for the taxpayer."

• David Cameron became embroiled after he incorrectly claimed that independent members of the Commons standards committee had the "casting vote" in deciding whether to censure Miller. No 10 admitted the prime minister had made a mistake. The committee consists of three lay members and 10 MPs.

Cameron called on the press to leave the matter alone. Speaking during a visit to Devon, he said: "What happened yesterday is that Maria Miller was actually cleared of the original charge made against her. It was found she had made mistakes, she accepted that, repaid the money, she apologised unreservedly to the House of Commons … So I think that we should leave it there."

• Labour sought to maintain the pressure with Thomas Docherty, the Dunfermline and West Fife MP, writing to the Metropolitan police calling for an investigation into Miller's expenses. Docherty wrote: "Given the widely differing conclusions of the commission and the committee regarding the serious allegations made about Mrs Miller and the fact that both the commission and committee feel that Mrs Miller did not co-operate with the inquiry, I believe this matter warrants further investigation and I believe the Metropolitan police are the appropriate body to carry out such an investigation."

The row between Miller and Hudson erupted at the end of November after the latter asked for details of the minister's mortgage dating back to the year she bought the Wimbledon property. Miller thought this was outside the terms of the inquiry because the period pre-dated her election to parliament in 2005 and marked a widening of the enquiry which she had been told had concluded in early October. Miller said the enquiry was meant to establish whether she benefited financially by allowing her parents to live in the home.

In a sign of her frustration Miller wrote to Hudson in December: "If you still wish to pursue the new matters which I consider are manifestly outside the terms of your investigation and beyond the scope of your enquiry, that would not be appropriate and I cannot agree to this unless it has been considered and formally sanctioned by the standards committee."A spokesperson for the culture secretary said: "The inquiry took 16 months during which time Maria was delivering major pieces of government policy, so of course she hoped it would be brought to a conclusion swiftly. In October 2013 the commissioner said she had all the information needed to complete her report.

"However, two months later, Maria was asked to provide further paperwork from nearly a decade before she was an MP, paperwork that did not relate to the central allegation – an allegation which has now been dismissed by the committee."

But Miller was offered some support after Kevin Barron, the chair of the House of Commons standards committee, and Hudson issued a joint statement to correct "misconceptions" in an attempt to show they were not at odds over the conduct of the culture secretary. The pair indicated that the minister had been asked to make a smaller repayment because MPs on the standards committee .

They said: "It should be noted that after the commissioner [Hudson] had concluded her inquiry the committee was able to secure further information from Mrs Miller on which to base its conclusions. The committee required Mrs Miller to give more information about her mortgage claims, and as a result it was revealed that Mrs Miller's mortgage had increased by over £150,000 after her election, not the £50,000 that the commissioner had discovered. The committee's calculations as to whether Mrs Miller's claims were justified were all based on this higher figure."