How to free up storage taken by the "On My iPad" app?
Well, it was sort of a mystery.

I'm not really sure which of my actions produced desired effect.


I copied a lot of files to iPad using the same "On My iPad" application to completely use up all available space. Then I deleted them via the Settings - Storage and also via the "On My iPad" app (where I could see the files going into the "Recently Deleted" location and where I could delete them completely).
I played with iTunes, connected the iPad to my Windows PC.
Rebooted iPad by holding the button on the top of the case and waiting till "Slide to turn off" appears on the screen.
Rebooted iPad by holding both the button on the top of the case and the Home button simultaneously till the screen goes black and then the Apple logo appears.
Also, probably, rebooting while it was connected to a PC with iTunes running.
Deleted the iPad's backup from the iCloud.
Turned on the iCloud Drive.


Eventually, I looked into the Recently Deleted location in the "On My iPad" application and I saw a bunch of files and folders that I have deleted few days ago. Those files that were hiding somewhere and taking up that 1.02 GB of space. As you can see from the screenshots in the question there was nothing showing up in the Recently Deleted location at the time when I was writing this question.

So, I don't really know what made those files to hide. I suspect that those files were not properly moved into the Recently Deleted area, because when I tried to delete them few days ago the iPad's storage was completely full and the deletion operation didn't go correctly. What an irony - there is no free space on the device, but when I try to delete a file it disappears from the view, but still stays taking up the space.

Most importantly, I don't know what made those files become visible again. Probably one of the things I did above.