Hey there Samfritch,
It sounds like you are deleting your Voice Mails, and they are reappearing and cannot be deleted. I recommend the troubleshooting from the article named:
iPhone: Troubleshooting voicemail
- Restart your iPhone.
- Check for a carrier settings update.
- Update your iPhone to the latest version of iOS.
- Make sure that your voice mailbox is set up properly. You can verify this by dialing your phone number from your iPhone.
- Make sure that you have a good cellular network connection. You can verify this by seeing whether you can make an outgoing call.
- Call your iPhone from another phone and try to leave yourself a voicemail.
- Reset your network settings by tapping Settings > General > Reset > Reset Network Settings.
- Contact your carrier and make sure your iPhone is properly configured with the voicemail feature on your plan.
Thank you for using Apple Support Communities.