The Bilsdale transmitter mast near Helmsley caught fire on August 10, and though the cause is unknown it left homes in north-eastern North Yorkshire, Teesside and County Durham without a TV and radio signal for several weeks.
Customers were advised by the BBC to use iPlayer and Sounds to watch and listen to programming online until the service was restored.
More than 400,000 households affected have now had their signal restored but others have not due to an issue over securing permission from the landowner to erect a temporary mast.
Refunds will be offered to customers unable to receive TV coverage for more than a month and who do not have cable or satellite platforms to enable them to watch iPlayer.
The Northern Echo newspaper, which covers County Durham and Teesside, campaigned for the refund on behalf of its readers.
Homes that had their signal restored within a month will not be entitled to the rebate.
The mast is operated by Arquiva, who are still in talks to erect a 260ft mast which is shorter in height than the damaged structure.
A TV Licensing spokesperson said: "Customers in the affected area who have been unable to receive TV coverage for over a month, and who are unable to view BBC programming through BBC iPlayer and on cable and satellite platforms, will be eligible for a refund or be offered a free extension to their TV Licence to cover the months affected.
"We are continuing to work with our suppliers Arqiva to ensure services in the affected area are resumed as soon as possible."
Licence extensions are also available for those affected.