I don't use Kerberos, but I'm sure some other folks around here could help you with your query.
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When configuring kerberos, you will need to have SETSPNs for all servers in the cluster (including Backup primary servers), and any load balancers or reverse proxies.
As the backup primary server isn't actually part of the cluster, you are going to have to edit the Kerberos Configuration Script and add it.
At the point of failover to the backup primary server, no additional changes should need to be made
All the best
Totally makes sense. Thanks for the reply.