To make it clear, the Relocated Item Folder showing after the macOS update is not related to the external hard drive. All of your data and files are carefully reviewed when you upgrade your Mac to make sure they are accurate, valid, untouched, and stored in the right places. A folder called Relocated Items is created if any of your files cannot be moved to their new locations.
When an external hard drive is not showing up on Mac, the connection, the drive format, the Mac software and hardware, hard drive errors, etc., all could be the culprit. You can try some basic checks to solve the issue, including checking the Finder preferences, restarting your Mac, reconnecting the drive, changing a USB port/cable, and checking the drive cable.
If the external hard drive is still not showing on the Desktop after you check to box in Finder preferences, does the drive appear in Finder's sidebar, and the Disk Utility? If the drive is greyed out in Disk Utility, try to mount it manually.
Have you used a third-party app to read-write the external hard drive, perhaps an NTFS for Mac driver? If so, you need to update it after the macOS update to make it compatible. If the current NTFS for Mac driver is not working, try iBoysoft NTFS for Mac, it is always timely updated to support the latest macOS.