Ministers have said they will redirect cash from wealthy areas to England’s most deprived town and cities, reversing a trend established during a decade of austerity in a shake-up of local government funding.

The government also said it would consider “on a case-by-case basis” requests from cash-strapped councils who wanted to balance their books by raising council tax above the current threshold of 5% for upper-tier authorities and 3% for districts.

Up to now, only councils that have formally declared themselves to be in effect bankrupt, such as Croydon and Thurrock, have been allowed to charge council tax above cap levels. Earlier this year a request by Medway and Somerset councils to charge 10% was refused.

A new £600m recovery grant will be targeted at England’s most deprived towns and cities, mainly in the north and Midlands, alongside longer-term “fair funding” plans intended to rebalance council funding away from the more affluent south-east.

The changes, announced on Thursday, were billed by ministers as an attempt to “fix the foundations” of local government and make funding more effective by ensuring councils with high needs in adult social care and child protection get a fairer share of funds.

Angela Rayner, the deputy prime minister and local government secretary, said: “For too long, councils have been let down by an outdated and inefficient funding system which has led to public services creaking and taxpayers’ money not being spent efficiently.

“Whilst there’s no magic wand to fix what we’ve inherited, we’re taking the necessary steps to fix the foundations of local government by creating a fairer system and ensuring every penny is spent on the services so many people rely on every day.”

The financial problems dogging councils have driven an unprecedented number of them into or close to insolvency in recent years as they struggle with soaring costs for social care, special educational needs and homelessness.

The Ministry of Housing, Communities and Local Government said in a statement: “The government is under no illusions about the scale of the issues facing local government, and this settlement will begin to address the pressures that councils are under. But this is a long-term project to deliver financial security for councils, and we recognise that we may see some continued instability as we adjust to the new system.”

The proposed new funding system from 2026-27 will be heavily weighted around deprivation and is likely to create winners and losers, triggering criticism from the Conservatives that it would result in funds being “reallocated to Labour strongholds’ and alarm among councils in rural and more affluent areas their own dire financial problems will be ignored.

Tim Oliver, the chair of the County Councils Network, said: “Whilst deprivation is a key indicator of a council’s need, it is not the only indicator nor the most important measure of financial distress. The reality is that it is demand and market failure across adult and children’s social care and special educational needs services that are pushing councils to the brink.”

An extra £250m for children’s social care, aimed at new initiatives to support families and reduce the numbers of children being taken into care will be introduced in April and targeted at councils where child protection services are under most pressure.

Plans to move to three-year funding agreements for councils, changes to business rates, and the removal of many of the grant funding pots introduced under last government will be brought forward to try to stabilise and simplify council finances.

David Phillips, of the Institute for Fiscal Studies, said: “The changes made for the coming year suggest these longer-term changes will also have an emphasis on redistributing funding to more deprived areas, to offset cuts they bore the brunt of during the 2010s. Whether that is fair will be in the eye of the beholder.”

Sir Stephen Houghton, the chair of the Special Interest Group of Municipal Authorities, said: “It is very welcome that the government have targeted new funding on a deprivation basis, recognising the disproportionate nature of the cuts and demand increases that the most deprived areas have seen over the last decade.”

Kevin Hollinrake, the shadow local government secretary, said: “They are also sticking two fingers up to rural areas through the scrapping of the rural services delivery grant and almost certainly reallocating it to Labour strongholds. If you thought Labour had finished taxing you, think again.”