Hello James and welcome to Seven Forums.
I'd start by safely removing the external Seagate from the Asus. If the computer is powered up and the safely remove icon is showing in the system tray, use that. Otherwise, power off the computer and then remove the Seagate just to make sure you're not interrupting any activity.
I'd probably try using a restore point and return the computer to a date/time prior to the backup problem. See if the backup works. System Restore
Whether the restore point works or not, I'd next check for any damaged or corrupt system files by running a system file checker scan from an elevated command prompt (option two.) If any problems are found, run the scan 3 times and reboot the computer after each scan. SFC /SCANNOW Command - System File Checker
Please post back if any of this helps or not.