[Ruth Levandowski]

My phone keeps popping up with "Apple ID Verification" asking to enter in a password for an email account in Settings (I don't have access to this)

[Tartarus]

My phone keeps popping up with "Apple ID Verification" asking to enter in a password for an email account in Settings (I don't have access to this)

Is there any apps in AppStore waiting to be updated?

Most like explanation is: At some point in the past you downloaded an app with the credentials of someone else and now in order to update that app, you’ll need the credentials from that person.

To get the latest version of that app anyway without those credentials is to delete that app from your iPhone and to download it using your own credentials.