Q: iTunes tells me my iPod classic is no longer authorised on my computer, how can I reauthorise it

iPods (that are not iPod touch) do not need to be authorized, so if you are getting an error message that says something about "no longer authorized" (when syncing iPod), it may be related to your current computer not being authorized with your Apple ID (iTunes Store account).  Therefore, you can no longer sync content purchased from the iTunes Store, IF that content uses DRM (copy protection). 

The command to Authorize This Computer is in iTunes, in the menu bar under Account -> Authorizations.

If the problem is something else, please post back with additional details about your situation.

An iPod (that is not an iPod touch) would not appear on a list of authorized devices for an Apple ID.  It's just the computer that syncs the iPod that needs to be authorized.  The iPod itself is not authorized with any Apple ID (except for iPod touch which is treated like an iPhone or iPad).

Is it possible you used a different Apple ID to purchase content (with DRM) that iTunes is attempting to sync to your iPod?  Like an old Apple ID, or an Apple ID of a family member with whom you share content.  If so, you need use the Authorize This Computer command again, and this time enter the old (or other) Apple ID and its password.  A computer can be authorized with multiple Apple IDs. 

If you're not sure, does the error message identify the songs (or other media) that cannot be synced?  If so, find one of those items in your iTunes library, right-click it, and Get Info.  In the Info window File tab, it should show you the Apple ID used to make the purchase (purchased by).