How will council cutbacks help the national debt (UK)?

As I understand it, local councils are funded by the people living in the area paying council tax. Those councils have been told by the Government to make cutbacks. If we as council tax payers are paying for those services that are being cut, it is not costing the Government anything, so they will not save anything by cutting them. As such, the council have two options. They can reduce what we pay in council tax to reflect the reduction in services, in which case, financially they will be no better off, or they can continue to charge us for services they are no longer providing, which surely must be fraud?