I want to make a purchase with my Apple gift card, but it asks for verification.

You may need to have a valid payment method on file, even if the app you want to download is free.

1.

See -> What you can buy with your Apple Gift Card or Apple Account balance

What you can buy with your Apple Gift Card or Apple Account balance – Apple Support (UK)

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Apple Gift Card and Apple Account balance can't be used for some purchases

• You can't use your Apple Account balance to send gifts or gift cards.
• If you use Family Sharing, your family members can't spend your Apple Account balance. Funds that you add to your Apple Account balance are only available to you. Find out more about how purchases are billed to your Apple Account balance.
• Some purchases might require you to have a credit card on file, even if you use your Apple Account balance to make the purchase.

2.

To 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.

See 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.

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 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.

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

If you change Country or Region, you must comply with all regulations described here ->  Change your Apple ID country or region

Change your Apple ID country or region - Apple Support (KW)

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What to do before you change your country or region

• Spend any balance remaining on your Apple Account balance. If you're not sure whether you have a balance, check your Apple Account balance.
• Cancel your subscriptions and wait until the end of the subscription period. 
• Wait for any memberships, pre-orders, movie rentals, or Season Passes to complete, and wait for any pending store credit refunds to process. You can cancel a pre-order.
• Make sure that you have a payment method for your new country or region. See what payment methods you can use.
• Redownload your apps, music, movies, TV shows, and books on any devices that you might use to play or read them in the future. Some types of content might not be available in your new country or region. Learn what types of content and services are available in your new country or region.

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You need to ZERO your Apple Account balance, it is depending on the amount:

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If your balance is less than the cost of a single item, contact Apple Support directly, they will help you clear your Apple Account balance: Get Support

2.

See -> Check your Apple Account balance

Check your Apple ID balance – Apple Support (UK)

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If you have Apple Account balance you can't use.

You can't redeem or return Apple Gift Cards, App Store & iTunes Gift Cards or unused Apple Account balance for cash, except as required by law. If your jurisdiction allows the refund of redeemed gift balances, you can contact Apple Support to request a refund. 

When you contact Apple, include the following information:

•           Your Apple ID.

•           The refund amount.

•           Your current delivery address. 

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You need to Cancel your subscriptions and wait until the end of the subscription period. 

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Make sure that you have a payment method for your new country or region. See what payment methods you can use.

You need to have a debit / credit card issued by a bank in your new country.

You need to have a billing address in your new country.

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