Are local councils broke?

According to the County Councils Network
Independent review of local government spending need and funding
An independent analysis of councils’ financial sustainability up to 2025 from PricewaterhouseCoopers LLP (PwC) stated that
Councils in England could face a funding black hole of over £50bn over the next six years, Read the full report on methodology and regional breakdowns to get the facts

this prediction has been echoed by the Local Government Association. In a June 2018 report it predicted a 7.8 billion funding gap by 2025.
More insight is given by their regular State of Local Government Finance Survey which has been every January since 2012 to coincide with councils setting their annual budgets. The results give a snapshot of the key pressures facing councils and the impact of ongoing financial uncertainty on their communities.

Other sources of information on this include IFS analysis of local government finance

they have also concluded that the present model of funding is unsustainable.

and a briefing document explaining the present situation

London Councils papers and news relating to local government funding reviews

UK government documents

House of Commons Library briefing paper 

Reviewing and reforming local government finance