Council funding has always come mainly from central government. Councils use the council taxes they raise to pay into a central pot and then redistribute via a funding formula based on perceived need. Wealthy areas receive back less than they contribute. Some areas can raise less because they have fewer high value properties and the formula is supposed to address that.
What this government has done since 2010 is adjust the formula, so that there is less redistribution. People blame their own councils, especially in Labour areas, when there is less money to spend. However, in reality, it is because the have been given less money from the central pot. It's been a way of "blaming" local councils and diverting from the role of central government. Meanwhile, thriving areas (mainly Conservative) are receiving more or, at least, not having their grants cut.