That is not likely to be a drive problem.
You generally see that message when you have linked the files on your Mac to your iCloud account, and are storing the Master copy of the file on iCloud. This feature is called "Optimize Storage".
But as you are now seeing, the setup has some unexpected issues. In addition, using Optimize storage is like an addiction, and is VERY hard to stop storing files on iCloud. [in my opinion] far superior long term results are obtained by adding and additional external drive, and moving some large items to that external drive. Like Music, videos, pictures, but NOT applications.
To be able to complete the backup, you Mac must spend enough 'awake but not too busy' time, logged in with the Apple-iD that owns those files, to fetch the Master copies of those files down from iCloud and allow the backup to complete.