Local authorities were unable to collect up to 40% of council tax due from low-income households that had the charge imposed on them for the first time last year.

Council tax – set on average at £5 a week – has been levied on the poorest households in England since April last year as part of a cut in benefits. But such is the squeeze on household budgets, say campaigners, that the poor cannot afford to pay even these small sums.

The result has been widespread non-payment. Nationally, more than a fifth of council tax charged to working-age claimants was unpaid at the end of 2013-14.

The figures, obtained from responses from 140 councils to Freedom of Information requests by the anti-cuts group False Economy, reveal that some of the biggest towns and cities were left chasing millions of pounds from the poor.

Liverpool collected 61% of council tax due from the poor, leaving the city short by £3.5m. In Birmingham, the non-payment rate among the vulnerable was 30%, leaving the council seeking to recover £3m in lost revenue. Leeds, Nottingham and Sheffield were all chasing more than £2m each in tax from those on the lowest incomes.

A report published last month by Child Poverty Action Group and the Zacchaeus 2000 Trust said almost 40% of Londoners affected by the cuts had been sent a court summons for council tax debts in 2013-14, with more than 15,000 claimants' debts referred to bailiffs.

In Haringey, north London, which collected 80% of the council tax due from benefit claimants, hundreds of households have been taken to court to recover unpaid tax – with non-payers threatened with bankruptcy, repossession and ultimately prison.

Last week, sitting in the magistrates court in Tottenham, Dick, 49, said there was "no way" he could afford the £7-a-week council tax his housing association two-bedroom flat was being charged. He has walked with a stick since his Achilles tendon snapped in 2012.

"I don't work. I get employment support allowance which is £70 a week and my son lives with me and he gets a few hours on a market stall. After rent and everything else we have about £140 a month to live on. Food, clothes, the lot. I go down the food bank to eat. Can't afford to heat up food because we cannot put money into the gas meter. How can I afford the council tax too? We never paid this before. It's just getting the poor to pay up. That's all it is."

Dick said he had offered to pay £3 a week towards council tax after working out his finances with the local Citizens Advice bureau, but the local authority did not respond to his offer. Instead the council has asked for the full year's council tax to be paid immediately – £350 – plus the cost of recovering his unpaid tax through a liability order of £125. "It's ridiculous. I worked all my life. Never needed anything. Now I got nothing they want to get that."

A spokesperson for False Economy called for the cuts to be reversed. "These figures show that people on low incomes are struggling to cope with council tax benefit cuts, just as the government was warned they would. Households are left either falling into debt and at risk of legal action, or taking money for food and essentials to plug the shortfall, in what is a government-created personal debt crisis."

Councils said they were caught in an "impossible situation" as ministers had forced local authorities to pass on £500m in cuts when the scheme was introduced – and there would be further reductions in the discounts the poor received as town hall budgets were squeezed in the coming years.

Sharon Taylor, chair of the Local Government Association's finance panel, said: "Councils would need to find £1bn by 2016 to protect discounts for those on low incomes.

"At a time when local government is already tackling £20bn worth of cuts, this is a stretch too far. Many councils have been put in an impossible position. No one wants to ask those on the lowest incomes to pay more. But pressure on funding for local services means many councils have had little choice but to reduce the discount."

Hilary Benn, the shadow cabinet member responsible for local government, said two million of the poorest people were affected by the council tax hikes. "These figures show that many of the people affected, including single parents and disabled people, are finding it very difficult to pay the Tories' tax increase. The government was warned that this was going to be Poll Tax mark two, and so it is proving."

The government defended its changes, saying it had "worked with councils to freeze council tax for the last four years" for most residents.

Kris Hopkins, the local government minister, said: "Our reforms to localise council tax support now give councils stronger incentives to support local firms, cut fraud, promote local enterprise and get people into work. We are ending Labour's something-for-nothing culture and making work pay."