Hi Notimewasted,
We see you're prompted to sign in with your Apple ID in order to download apps, however when attempting to enter your password to do so it loops back to the original message, is that correct?
If so, we'd recommend going to Settings > [your name] and tap on "Media & Purchases". If you're presented with an option to sign out, do so. Then sign back in.
After that, attempt to to download an app from the App Store again and complete the verification process if prompted. If you're still looped back around to the same message, are you prompted to enter payment or billing information during the verification process? If so, you can see how to edit your billing info here:
Cheers!