Usually, yes: Council Tax Support is run by your local council, and a claim or award normally does not transfer automatically when you move to a different area. Tell your old council that you have moved and contact the council for your new address to check its scheme and apply as soon as possible, as backdating rules vary. You may also need to provide your new tenancy details, income and household information. Check the relevant council’s website or GOV.UK for the application process and deadlines.