Can an app keep onto its root access permissions after the phone is turned off? My question is plain and simple. Can an app keep onto its root access permissions after the phone is turned off? Is there a way an app with root access can keep this permission and start running with this permission without asking for it again once the device is turned back on?

For all I know, it can not, but it can be set to "grant root access whenever it demands" by superuser app.

The reason I ask this question is: I talked to the support service of the spy apps such as mSpy or FlexiSpy (apps that fetch your every single info but requires root access) which accept their app would not survive a factory reset but insist that their app would keep working properly after a reboot.
Apps like SuperSU will remember your decision regarding root access to an app and will give that reply each time the app asks, unless you specify otherwise in the settings.
Make sure your app starts after the device boots.

If your application needs root permissions during boot (not after the device has booted) you'll need to enable "su during boot" (SuperSU)