The correct procedure to restore an AEM instance backup depends on your backup strategy.
You may want to check the official documentation on backup and restore for AEM.
There are different common approaches to backups in AEM
There is another approach to restores when it comes to publish instances:
Usually, you will have multiple publish instances available. If you only need a restore for one of them, you can also create a clone of one of the others and restore that to the desired host/instance. You would need to adjust certain configs on the clone to reflect the target instance (probably replication agents for dispatcher flush and similar). In my eyes, this is actually the easiest way to restore a single publish instance (or to add additional ones for scaling purposes).
Hope that helps!
Without any further knowledge about the situation it's hard to give a clear recommendation.
There are ways to address and potentially fix a repository corruption. Please refer to the following articles for guidance:
If you are able to fix the corruption following the described actions, you're fine.
Of not, you may have to follow one of the restore approaches as outlined in my initial response.
In addition to that, I would also recommend to perform an analysis to see if you can find the root cause for the corruption to avoid similar situations in the future.
Hope that help!
The second error about AuthenticationSupport is caused by the first one about missing segments.You can ignore that one for the moment and focus on fixing the repository issue. So you should check if the linked KB articles help you resolving the segment store corruption.
If that does not help, you will need to follow the restore approach.
Have you already been able to identify the cause for the repository corruption?
Have any files been deleted on the file system?