A council says it needs to postpone contributions to staff pensions due to a prediction its budget shortfall will treble to £33m next year.
Peterborough City Council says paying for Covid-19 related services has contributed to a budget gap which is already £11.8m this year.
The Conservative-led council said it also needed a "holiday" from loan repayments to government.
The government said the council had had £15m emergency funding.
Peterborough's Conservative deputy council leader Wayne Fitzgerald told the BBC's Politics East TV programme that councils had "borne the brunt" of work to control the spread of coronavirus.
"This is a strong business case - it's not a begging bowl," he said.
Meetings with MPs
He said the council had to pay for things such as anti-infection measures in care homes, while losing income from things like car parking and museum entry fees after lockdown started.
Mr Fitzgerald said the council's total annual budget was about £440m and extra government funds to pay for the extra services would run at the end of September.
The council said it was also going to ask the government for a "holiday" on loan repayments on money borrowed from the government's Public Works Loan Board, but would continue to pay interest.
Informal talks have been held with the city's Conservative MPs Paul Bristow and Shailesh Vara and with the Ministry for Housing, Communities and Local Government (MHCLG) and a formal request for more financial support was expected to be made by the end of this month, Mr Fitzgerald.
In a statement, the MHCLG said Peterborough had received almost £15m emergency funding on top of a pre-pandemic increase in spending of £9.5m a year.
The government has announced it would introduce a compensation scheme, providing councils with 75p in the pound for income lost in areas such as parking charges.
Mr Fitzgerald said that scheme would amount to about £2m and would apply to this year's budget, so it would not affect the projected £33m shortfall next year.
Politics East is on BBC One at 10:00 BST on Sunday, and also available on the BBC iPlayer after broadcast.