glacial Time Machine backup

My Time Machine is slow to back up again. For the past history, see https://discussions.apple.com/thread/8240655.

A week ago I tried again to back up this Mac Mini. I have a Hitachi one-terabyte external hard drive plugged in by USB 2.0. Each day for the last week I have had the same experience: Time Machine spends approximately six hours “preparing backup.” Time Machine then starts the back-up process. After another three hours Time Machine has backed up approximately 1.6 gigabytes of data. That is nowhere near enough.

Between the two computers that I back up, this Hitachi hard drive should be full. Nonetheless, my understanding is that Time Machine should delete the oldest backups in order to make room for the latest. That would be fine with me.

I admit that for the first six days I may have made an error. I turn off the Mini every night. For the first six days I shut down the Mini without my taking any specific action as to Time Machine. In contrast last night, after reading some stuff, I clicked “Skip This Backup.” I waited for Time Machine to stop. When it did, then I shut down the Mini.

Doing the “Skip This Backup” appears to have had no effect. Time Machine is still, as I write this, “preparing backup” after two hours.

Yesterday, I ran Disk Utility on the destination Hitachi. Disk Utility reported no problem with that hard drive.

What is distinctive this time is a mysterious folder that appears in Trash:

This folder appeared in Trash last week after I started Time Machine. As long as the external hard drive is connected to the Mini, I cannot delete this folder.

This morning I did stop Time Machine and disconnect the external hard drive. I was then able to delete the mystery folder. After I reconnected the external hard drive and restarted Time Machine, the mystery folder reappeared in Trash. I do not know the significance of the blank folder.