What causes Time Machine to perform a non-manual verification?
I have not witnessed a Time Machine verify process in many years. If you have your Time Machine icon showing in the menu bar, you can click it while holding the Option key and there's a Verify Backups option.  I've never had this run on its own unless the disk I was using for Time Machine backups was potentially faulty.  In which case, I would run Disk Utility Verify/Repair on it and watch out for any S.M.A.R.T errors being reported. Then I would manually kick off the Time Machine verification. If the disk reports errors in S.M.A.R.T then the disk is dying and urgently needs to be replaced.
The only other time I have seen verifications is when you mount a DMG or Sparse Image the Finder will attempt to verify the integrity of the image(s).  This can take a long time because it has to scan every bit of the file and these backups can be quite large.
I have been using Time Machine over a network to a NAS where the Time Machine data is stored within a disk image called a sparse image.  That means it's one large directory where the data within is split into bands with binary chunks of 134MB each.  If it wasn't a sparse image it would be very similar to a DMG disk image single file with a large fixed size.
One way to bypass the verification step is to mount the disk from the Terminal command line instead of using Finder. If the file resides on network volume, you need to mount the share first either via Finder or command line. You can eject the disk from Finder as normal after mounting from the command line.
hdiutil attach path/to/image/filename.sparsebundle

It should then be mounted under /Volumes and otherwise accessible in the Finder.
If you are mounting from an Apple Time Capsule, then the disk might be failing. If you are sharing this Time Machine backup on a NAS the file might have been damaged in some way. Causing Time Machine to detect a problem that needs to be repaired.  If it's a soft error then it can be repaired. If it's hardware failure like bad sectors on the disk then you need to replace the disk.
It would be strongly recommended that you check the integrity of the storage medium. i.e. Disk and that it's not having problems indicating an imminent failure.