Restore of VM check SKU of existing Public IP and provide same SKU for new PIP - availability zones
Restore of a vm to new vnet/subnet will not always get a public ip. If the vm that you want to restore has public IP SKU with Standard or higher the process do not check for that and will fail provisioning a public ip.
Issue happens if you restore a vm that use availability zone or Network Load Balancing in front of vm.
Please have restore job check SKU for Public IP of VM prior to restore and deploy VM with Public IP with same SKU. Then the restore will complete. Please note that you can assign public IP either via PS or portal. Just remember to provide same SKU or it will fail.