I am unable to add a New Payment Method to my Apple ID

Do you use VPN? If so, delete the VPN profile in Settings/General/VPN & Device Management/VPN. Don't just turn it off; delete the profile.

Not all cards can be used with the App Store, that’s a choice of the card issuer. So start by checking with the issuer to see if they accept cards for use in the App Store. Are you getting a “declined” message? A “card declined" message means that the issuer of the credit or debit card you have on file in your iTunes account rejected a charge or account confirmation, or the confirmation never got to the issuer. It is not an Apple problem; Apple is just passing through the message from your issuer. There are a number of reasons the charge or verification might be declined:

• Apple is not allowed to accept credit or debit cards in your country
• The card is not recognized in your country
• Insufficient balance or account in arrears
• Account suspended for suspected fraud (not necessarily by you)
• Error in card number
• Wrong expiration date
• Wrong security (CVV) code
• Address in your iTunes account does not match exactly the billing address for the card
• If you are trying to add the card to the Apple Wallet the issuer doesn't support the Wallet
• The card was not issued in the region of the iTunes store
• You are connected through a VPN provider that does not terminate near your actual location.

Here is additional information from Apple→If your payment method is declined in the App Store or iTunes Store - Apple Support

Related issues for specific messages:

• If a message says 'billing problem with previous purchase' or 'verification required'
• Your account has been disabled in the App Store and iTunes