It won’t let me add a new payment, Why?

See -> If you need to change or update your Apple ID payment method

If you need to change or update your Apple ID payment method - Apple Support

• If you want to change your card, you need to add first the new card, then remove the old card.
• or if you use your Apple ID balance, set your Apple ID balance as your default payment.

or

Add a payment method to your Apple ID

Add a payment method to your Apple ID – Apple Support (UK)

• If you want to change your card, you need to add first the new card, then remove the old card.
• or if you use your Apple ID balance, set your Apple ID balance as your default payment.

See also the following Apple articles:

Payment methods you can use with your Apple ID

Payment methods you can use with your Apple ID – Apple Support (UK)

fyi: Not all credit & debit cards seem to work with App Store / iTunes Store, try a different one, or preferably a credit card.

If your card is not accepted as a paymenf method, you need to contact your bank.

Remove a payment method from your Apple ID -> Remove a payment method from your Apple ID - Apple Support

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If you can't remove a payment method

If you have subscriptions like iCloud+, have an unpaid balance, or you use purchase sharing, you must keep at least one payment method on file.

• Do you have a subscription? You can cancel iCloud+ and cancel other subscriptions from Apple. When your subscription period is over, remove your payment method.
• If you're in a Family Sharing group and you use purchase sharing, the family organizer must have a payment method on file. If you're the family organizer, turn off purchase sharing, then remove the payment method.
• If you have an unpaid balance, you might not be able to remove a payment method. Instead, change your payment method.
• If you still can't remove a payment method, contact Apple Support.

If you have followed all the steps and instructions and if you still need help, get support from Apple directly -> Get Support or call Apple in your region see -> Contact Apple for support and service or check -> here for more support and assistance.

Does it say why it won't? Can you provide the exact wording of error messages?

See this article about updating payment methods on various platforms (iPhone, Mac, etc.): If you need to change or update your Apple ID payment method - Apple Support

"If you change your address, the credit or debit card information that you use with your Apple ID might be automatically updated by your financial institution, and you don't need to take action. But if there's an issue [such as you having a subscription or being a Family Sharing Organizer], add a new card first, then remove the old card."

Here is how you update your payment method on an iPhone or iPad:

1. Open the Settings app.

2. Tap your name.

3. Tap Payment & Shipping. You might be asked to sign in with your Apple ID.

4. Tap Add Payment Method, enter the new payment method, then tap Done.

5. Tap Edit.

6. Tap the red delete button next to the old payment method, then tap Delete.