Hello IOTBNTBP1,
Firstly, as part of the backup and restore operations, the following roles are assigned to the AKS cluster, Backup Extension Identity, and Backup vault.
So, add the Storage Blob Data Contributor role at Extension Identity, and the Storage Blob Data Reader role at Backup vault. Refer the required roles and permissions : https://learn.microsoft.com/en-us/azure/backup/azure-kubernetes-service-cluster-backup-concept#required-roles-and-permissions
Then, yes since the storage account allows Private access, the backup vault needs to be added as trusted access in Storage Account Network Settings.
Hence, please add it and save the settings as follows,
Post these changes, monitor the backups and check if he still encounters the error or not.
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Thanks