Yes, definitely! When moving to a new district, you have to notify your current local authority that you are moving out of the district. You can de-register from your local authority at your local residents registration office. The same applies for registering at your new local residents registration office. You will have 14 days to de-register and register. Please contact your local authority for information on the documents you will need to de-register and register.
By the way: you will still have to register your new address at your local residents registration office even if you are not moving to a different district.