My Mac osx10.8.5 desktop no longer backs up to Time Machine/Seagate 4TB external Disc
It makes the daily attempt to update, but gives an alert in a small window that the backup failed.
Earlier Mac models
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Launch Disk Utility and run the First Aid routine on the external Time Machine drive. With luck the problem will be something repairable.
Please look for guidance here: Repair a disk using Disk Utility on Mac - Apple Support