Your payment authorization failed. Please verify your information and try again, or try another payment method.

the apple staff said they've met lots of this issues previously, could anyone report this case to the company,

If so, they could have reported it. After all, they are closer to the source than we are.

We are also users of Apple products & services like you.

I can only add my payment method through itunes, but it didn't work on appleid.com.

Try again, many users have succeeded.

Get more help and assistant from Apple directly, see Contact Apple for support and service -> Contact Apple for support and service – Apple Support (UK) 

but I found I can not purches any product by using my account balance.

See -> Shopping Help - Payment & Pricing

Payment, Financing, Refunds and VAT - Shopping Help - Apple

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Using more than one payment method

You can also combine payment methods to pay for your purchase. Choose from the following combinations of credit cards and gift cards:

• Credit card(s) - up to two
• Apple Gift Card(s) - up to eight
• Apple Gift Card(s) - up to eight + one credit card
• Apple Gift Card(s) - up to eight + Apple Account Balance
• Apple Gift Card(s) - up to eight + Apple Account Balance + one credit card
• Apple Gift Card(s) - up to eight + Apple Pay
• Apple Gift Card(s) - up to eight + Apple Account Balance + Apple Pay
• Apple Account Balance + one credit card
• Apple Account Balance + Apple Pay

Please note that you can use the value from all types and generations of Apple gift cards. For more details, see: 

• How to use Apple Gift Cards and App Store & iTunes Gift Cards
• How to use Apple Store Gift Cards

During checkout, if you are paying with a gift card issued by a credit card company, you can combine it with one credit card.

Credit Cards or Debit Cards

Troubleshooting credit card & debit card payment errors

• If your card is new or recently reissued, make sure it has been activated.
• Double check the card type (for example, American Express), the account number and expiration date.
• Remove dashes or spaces from the account number.
• If everything seems to be correct, call the card issuer's customer service 800 number on the back of your card. Your credit card company may need to speak with you before they authorize your purchase.

Please call 1-800-MY-APPLE and speak to an Apple Specialist for more information.

and the customer service said they failed to verify my account, is there anything wrong I did?

Verify your payment information, you need to -> Sign in to your Apple ID account page and edit / update your Apple ID details -> https://appleid.apple.com                                                     

• On the Sidebar click Payment methods -> then tap Payments Methods and tap Manage payment method.

Or see -> 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 Account balance, you need to set your Apple Account balance as your default payment.

If you update your payment method or add funds to your Apple Account balance, It would take about 1 to 3 business days in order to process.

If you're still unable to update your payment method, Contact Apple Support.

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.

Get more help and assistant from Apple directly, see Contact Apple for support and service -> Contact Apple for support and service – Apple Support (UK) 

'Cannot Add Payment Method Contact Apple Support for more information.'

1.

See -> 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.

2.

See -> If your payment method is declined in the App Store or iTunes Store

If your payment method is declined in the App Store or iTunes Store - Apple Support

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How to complete your purchase if a payment failed or was declined

1. On your iPhone or iPad, open the Settings app.
2. Tap your name.
3. Tap Payment & Shipping.
4. Add a different payment method and remove the old one. 
5. Try to complete your purchase.

If you still can't make a purchase

• You might have an unpaid order. Learn how to pay an unpaid order.
• Contact your financial institution. 
  • You might need to ask your financial institution to allow international purchases on your debit or credit card.
  • If you want to know why a debit card, credit card, or other payment method was declined, speak with your financial institution. Apple doesn't have information about why the payment method was declined.
• In some countries and regions, your payment method might be declined because of changes to local law. Learn about billing and purchases in Europe or what to do if your card is declined in India.
• Still need help? Contact Apple Support.

2 different credit cards from Amex and Discover, and both of them failed to be added as the payment method

Then we go back to the first hyperlink, see -> Shopping Help - Payment & Pricing

Payment, Financing, Refunds and VAT - Shopping Help - Apple

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Security codes

The card security code is a unique three or four digit number printed on the front (American Express) or back (Discover/MasterCard/Visa/) of your card, in addition to the account number.

American Express:

Discover/MasterCard/Visa:

Daily spending limits for Debit Cards

Most debit and check cards have daily spending limits that may substantially delay the processing of an order, even if there is enough money in an account. When using a debit card, funds are immediately reserved in your bank account at the time you place your order. Please consult your card-issuing bank for information about your daily spending limits if you are having trouble placing your order.

When Apple charges your account

When you place an order, Apple will contact your credit or debit card's issuing bank to confirm it is a valid account. Apple requests an authorization, which holds the funds for the full or partial purchase amount. As Apple prepares your items, authorized funds will be unavailable and may display as "pending" on your credit or debit card's online statement.

If you are shipping an item, your card will only be charged after the item has shipped. For pickup items, you will be billed before your item is ready for pick up. If your order contains multiple items, you may see multiple charges on your credit or debit card statement. 

If you cancel your order before your items are billed, any authorizations placed on your credit card account can take up to 30 days to expire and any authorizations placed on your debit card account can take up to 10 days to expire. Contact your financial institution for their specific timeframes.

Credit cards accepted

In the U.S., we accept:

• American Express
• Discover 
• MasterCard
• UnionPay
• Visa

If you are paying for your order with an international Visa, MasterCard, American Express, or UnionPay credit card, please note that the purchase price may fluctuate with exchange rates. In addition, your bank or credit card issuer may also charge you foreign conversion charges and fees, which may also increase the overall cost of your purchase. Please contact your bank or credit card issuer regarding these fees.

Any gift card issued by a credit card company can be used as a method of payment on the Apple Online Store. During checkout, if you are paying with a gift card issued by a credit card company, you can combine it with one credit card.

You need to call -> 1-800-MY-APPLE and speak to an Apple Specialist for more information.

I successfully login my appleid account -> 👍

I can remove card from the payment method, -> 👍

but I can not add a new card

Please check -> 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.

'Your Apple Account Balance can’t be used with PayPal

(see what posted) In the last sentence of -> Using more than one payment method

• Apple Account Balance + Apple Pay

Please call 1-800-MY-APPLE and speak to an Apple Specialist for more information.