The chancellor, Philip Hammond, faces a chorus of calls to scrap the cap on housing benefit in his budget next month as a leading charity and local government leaders criticised government plans to fix the “broken” housing market as inadequate.

In her party conference speech, Theresa May promised to dedicate her premiership to addressing the shortage of housing, saying the state must get “back in the business” of building subsidised rented homes for those not able to buy. She announced a £2bn injection of public funds which the government said could pay for an extra 25,000 homes for social rent by 2021.

But the homeless charity Shelter said that, while the extra money was welcome, it was a tiny proportion of what was required and would not help about a million private renters who were in immediate danger of being made homeless as a result of the housing benefit cap.

Polly Neate, Shelter’s chief executive, said: “The money put forward is only a fraction of what’s needed, given just how bad this crisis has become. Building new homes also takes time, and time is not on the side of the million private renters who could risk being tipped into homelessness by the freeze on housing benefit. Whether a struggling family or a young person in low-paid work, the freeze is stripping away the help people desperately need to pay their rent.

“Given the tide of despair faced by hard-up renters, we are urging the government to abandon the freeze on housing benefit in the autumn budget or risk making more people homeless.”

A survey by Shelter of 3,978 renters found that 79% of those who were in work while also claiming housing benefit were struggling with their rent payments. More than half of these “working renters” on housing benefit were worried about losing their homes, while 71% said it was harder to find a decent affordable home now compared with five years ago.

The Local Government Association, equally concerned that rising rents have left many people who claim housing benefit with large shortfalls, has also called on the government to scrap the cap on housing benefit, which under current plans is due to carry on until 2020.

Separately, more than 50 leaders of Labour councils and Labour mayors have written to May demanding more funding so that councils can build the homes their communities need.

In their letter, the local Labour leaders say: “Your conference announcement follows in a long line of policy decisions that Tory ministers have made on housing since 2010, which has done nothing to fix the housing crisis, and in many cases made the problems worse.”

“Home-ownership across the country is down sharply, with almost 200,000 fewer homeowners since 2010, rough sleeping has more than doubled, private rents have risen faster than incomes, housing benefit spending has increased, and affordable housebuilding last year was at the lowest level in 24 years.”