Local government secretary Robert Jenrick has confirmed allocations for individual councils from the £1bn of extra funding announced by Prime Minister Boris Johnson.
All councils will receive a minimum of £100,000 from the fund, which has not been ringfenced.
But finance spokesman for the County Councils' Network, Cllr Carl Les, said the allocations were 'disappointing'.
He added: 'Government must now consider providing further funding to support councils who will now face a much larger shortfall than they anticipated.'
A new £100m fund has also been created to support local authority leisure centres during the crisis.
Culture secretary Oliver Dowden said: ‘It's vital that we keep ourselves fighting fit through the winter months and local authority leisure centres are crucial to this.
'This £100m fund will help keep leisure centres across the country open.'
Chair of the Local Government Association’s resources board, Cllr Richard Watts, said: 'Social distancing means leisure services will continue to face a challenging financial position for the foreseeable future.
'That is why it is also vital the Government’s Spending Review provides further investment so councils can protect and improve local services both this and next year, including leisure facilities.'