Leaders from the larger local authorities fear the second round of emergency funding will still fail to cover their costs during the pandemic.
Whitehall has confirmed how it will allocate the second £1.6bn package to support councils.
Smaller district and borough councils will receive a greater proportion of the funding this time around.
But council bosses faced with a rising social care bill say the fund does not address their pressures.
One leader in the North East told the Local Democracy Reporting Service the area had been left "short-changed" by the latest round.
Newcastle City Council has spent an estimated £60m tackling Covid-19, but has received £18.6m in support.
Council leader Nick Forbes said councils have been hit by a loss of income from parking, fines and taxes.
He said: "While the next round of government funding is welcome, Newcastle and the whole of the North East, has once again been short-changed.
"The total pot of monies made available is the same as the first funding round, yet we have received £2m less."
Leaders across England have largely welcomed the funding, which sees a greater proportion allocated to district and borough councils.
But upper tier authorities, such as unitary and county councils, say the rising cost of adult care and providing equipment for carers has not been addressed in the latest round.
Bristol's mayor Councillor Marvin Rees has urged the government to issue local authorities with an NHS-style bailout by wiping debts.
He said the the latest £26.4m would not cover the city council's £29m cost of coping with the emergency, such as paying for adult social care.
"The inadequacy of government finance and the drive of national government to control everything from Whitehall is causing major problems in our ability to respond to the crisis," the mayor said.
"What the Government is not doing is taking into account the lost revenue to local government.
"They are not grappling with the importance of local authorities raising revenue to cover the cost of core services now."
Councils have not been restricted in how they spend the money.
Most have so far been using the previous round of funds to cover adult social care, tackle homelessness and to protect vulnerable children in care, the government says.
The new funding is expected to arrive in their bank accounts in May.
But the Local Government Association (LGA) says Whitehall will need to provide "ongoing and consistent funding" to meet the extra costs local authorities are facing.
'Much more exposed'
Smaller district and borough councils, which look after things like bin collections and parks maintenance, received a big increase in the second round of funding.
Under the first £1.6bn, their share was about £10m, whereas under the second batch they will share about £214m.
David Phillips, from the independent Institute for Fiscal Studies (IFS), said this was welcome as district councils are "particularly reliant on the income they can receive from fees and charges for services, such as parking or leisure".
He added: "That makes them much more exposed to the effects of the lockdown."
On average, shire district councils got about 45p per resident in the first round but many got about £10 per person in the second.
The government has also announced it will delay plans to change the model by which councils are funded by a year.
In 2021, it had proposed authorities would fund services by retaining 75 per cent of business rates in their area, relying even less on a central grant from Whitehall.
Chairman of the District Councils Network Councillor John Fuller said he was pleased the government recognised the "incredible work" district councils do in supporting families and businesses.
The total support across both tranches, works out at about £57 per person when district and county councils are combined, according to analysis by the BBC.
Fire authorities and police and fire commissioners have been excluded from the analysis, but between them these 30 authorities have received £35m from the two rounds.
Reporting team: Paul Lynch, Daniel Wainwright and the Local Democracy Reporting Service