The first error appearing the log was:
The Health Service has downloaded secure configuration for management group "[Management_Group_Name]", and processing the configuration failed with error code Cannot find the certificate and private key for decryption.
The next error was:
Received configuratiion cannot be processed. Management group "[Management_Group_Name]". The error is Cannot find the certificate and private key for decryption
These events also conincided with an alert being logged in the Operations Manager Console: "Secure Storage Configuration Error".
For this particular situation, the fix was to remove a registry key. I deleted the value (not the entire key, just the default value) for the registry entry:HKLM\SYSTEM\CurrentControlSet\Services\HealthService\Parameters\SecureStorageManager
Once deleted, restart the OpsMgr Health Service.
My experience with this was that the error did not clear straight away, and the problem was not visible fixed immediatly. However, over a period of about 45mins, the SCOM server cleared itself, and returned back to a healthy state. If you are able to do so, a full reboot instead of just restarting the health service wouldn't hurt.