Highways authorities are defined in the  Highways Act 1980 .  In summary they are :

The Minister (presumably the Secretary of State for Transport) for motorways (special roads) and certain trunk roads (But after April 2015 Highways England)
The County Councils and Metropolitan districts
Transport for London for GLA  roads
The London Boroughs for roads which are not GLA roads

The Highways  Act does not deal with who is responsible for cleaning roads.  This is the province of  the Environmental Protection Act.   This post has been included as references are made to “highways authorities” in the EPA.