The watchdog introduced last year to govern MPs' expenses today hit back against threats that it could be reformed or even scrapped with a poll reminding parliament of the depth of public mistrust they still face in the wake of last year's expenses scandal.

David Cameron issued a warning before Christmas that the Independent Parliamentary Standards Authority should reform and improve the way it works after complaints from every corner of the House of Commons of slow payments, troublesome IT systems and unfair rules. He set an April deadline for Ipsa to improve or face being forced to change, meaning it could even be scrapped.

Today Ipsa published a poll designed to remind MPs that the expenses crisis is still fresh in the public's mind – with the majority professing to have little or no trust in their MPs to fairly claim taxpayers' money – in an assertion of its independence. The YouGov poll of 2,600 people found that 28% of people don't trust MPs to make legitimate expenses claims "at all", 35% trust them "not very much" while 23% have trust in them "to some extent" and only 7% a "great deal".

Sir Ian Kennedy, the chair of Ipsa, insisted that the system was working as he launched a long-planned formal consultation on how to improve the system, appealing for the public to take part in the process and making it clear they would not just listen to disgruntled MPs.

Kennedy said: "Ipsa was created by parliament to bring an end to the old, discredited expenses system and has introduced an integrity and probity to MPs' expenses. Our new approach is working, but I know it is not perfect. We have learned many lessons in the eight months that we have been regulating MPs' expenses and have taken them on board and acted where appropriate. But, of course, some still have concerns.

"We want to hear these concerns and the evidence which lies behind them. We want to hear not only from MPs but from taxpayers around the country. After all, it is the taxpayer who pays the expenses."

The launch of the consultation prompted a new round of criticisms from MPs. Tory Roger Gale called on Kennedy to resign and Liberal Democrat Bob Russell said that the organisation was the worst he had dealt with in 40 years in public life.

A consultation document launched today suggests that Ipsa is preparing to make some major concessions that could see MPs allowed to claim for mortgages again, more expenses paid to subsidise the families of MPs who need second homes to carry out their jobs and more money granted to fund constituency offices.

There has been particular concern that the rules, introduced after the election to replace the discredited previous regime which saw many MPs making extravagant claims, have proved more costly. Previously, non-London MPs could claim for interest on a mortgage on a second home in an arrangement that was severely criticised after some "flipped" the accommodation they designated as a second home in order to maximise their claims and which allowed them to keep the profit from those homes when they subsequently sold. That was replaced with a £1,400 capped rent, but there have been concerns that this has proved more expensive than the former interest payments.

Sir George Young, the leader of the Commons, indicated that the government expects reforms to the Ipsa scheme, while insisting that it still supports the independence of Ipsa. "We welcome today's decision by Ipsa to conduct a wide-ranging review of the expenses system, which will give them the opportunity to engage with both the public and MPs," he said.

"The expenses system is there to give MPs the support they need to carry out their jobs and Ipsa should look again at areas where the current scheme is not adequately doing that.

"I intend to submit evidence to the review on behalf of the government and am encouraging colleagues who are experiencing difficulties to do the same. We support the principles of independent and transparent regulation of MPs' expenses."