County councils in England have warned they may have to suspend non-essential spending because of what they say is a £600m shortfall in their budgets.
They said they faced tough choices after getting less than expected in an emergency government coronavirus fund.
They suggested ministers had favoured smaller district councils despite the fact they and unitary authorities were responsible for social care provision.
Ministers said the £1.6bn was being shared out "as fairly as possible".
And the body which represents district councils said they welcomed the additional money, which they said was a recognition of the "incredible work" they were doing locally in response to the pandemic.
Prior to Tuesday's announcement, county councils and unitary authorities had called for their social care budgets to be prioritised and warned of "unpalatable" decisions if an estimated £1bn in extra care costs caused by the virus were not met by central government.
The £1.6bn fund - the second tranche of a £3.2bn package of financial support - is to be allocated on a per capita basis related to the number of people living in each area, and an assessment of local authorities' individual needs.
County councils say they were "disappointed" that their share of the pot was, on average, 29% lower than what they got in the first round of funding announced earlier this month and it could have serious repercussions.
The County Councils Network, which represents 36 county councils and unitary authorities, warned of an immediate £600m funding shortfall due to extra demand for their services and reduced income from fees, charges and other streams.
A further £2.4bn could also potentially be lost due to a likely collapse in business rate revenues and non-payment of council tax, it added.
The organisation's chair David Williams said that unless further support was provided to all councils "some may feel that they have no choice but to suspend all non-essential expenditure, hampering our national efforts to defeat the coronavirus and prepare for the economic recovery."
He added: "With district councils receiving 35% of the new resources in shire counties, despite representing 10% of service expenditure in two-tier areas, it is clear that government is providing only one tier of local government with support for the immense pressure that lost income is placing on all our budgets.
"Unless a solution is forthcoming to address these growing additional pressures as the crisis goes on, and, crucially, our lost income, it is inevitable that the resources in shire counties made available to life-critical social care services will be reduced, resulting in extremely difficult choices facing our member councils."
And Sigoma, which represents urban unitary councils outside London, said the outcome was "very disappointing".
"The impact of Covid-19 is not being felt on a per-capita basis and funding should be allocated on the immediate pressures faced by social care authorities," it said.
In contrast, the District Councils Network welcomed the settlement, which will see the majority of the 192 district councils in England receive more than £1m each.
"This shows government recognises the incredible work of districts in supporting every family and business and the pivotal role they can play combating this virus and leading the local recovery," said its chair John Fuller.
Ministers say they have acted swiftly and decisively to help councils deal with the crisis, including bringing forward £850m in social care grants and allowing town halls to defer £2.6bn in business rate payments.
To further ease the pressure on councils, a review of their current resources and future needs designed to determine changes to baseline funding next year will not go ahead.
Neither will a shake-up of business rates which would have seen the amount of rate income councils retain from High Street shops - many of which are now closed - rising from 50% to 75% in return for grants being phased out.
Council leaders have warned that billions more in financial support may be needed to avert potential bankruptcies and have called for all losses of income due to the pandemic to be under-written by government.
Communities Secretary Robert Jenrick said the latest tranche of funding was being distributed "in the fairest way possible, recognising the latest and best assessment of the pressures they face".
"Councils are playing a central role in our national fight against coronavirus and the government continues to back them at this challenging time," he added.