I’m trying to download a free app to my phone and it wants me to add a payment method to my Apple ID Can I continue without adding a payment method to my Apple ID

If you want to make purchases or update / download apps, including free apps, you need to have a payment method on file.

If you don't have a credit card or don't want to use a credit card, you can also use your Apple ID balance as payment method, see -> Add money to your Apple ID balance

Add money to your Apple ID balance – Apple Support (UK)

See also: 

What you can buy with your Apple Gift Card or Apple ID balance

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

How to redeem your Apple Gift Card or App Store & iTunes gift card

How to redeem your Apple Gift Card or App Store & iTunes gift card – Apple Support (UK)

When redeeming a gift card, (1) the amount will be credited to your Apple ID balance and (2) then you need to set your Apple ID balance as your default payment.

(1) See -> Check your Apple ID balance

Check your Apple ID balance – Apple Support (UK)

(2) See -> Add a payment method to your Apple ID

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

• if you use your Apple ID balance, you need to set your Apple ID balance as your default payment.

If you update your payment method or add funds to your Apple ID 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.

Also read the articles below ⤵️

If you can't redeem your Apple Gift Card or App Store & iTunes Gift Card

If you can't redeem your Apple Gift Card or App Store & iTunes Gift Card - Apple Support

• You can't redeem Apple Gift Cards or App Store & iTunes Gift Cards outside the country or region of purchase. For example, if you bought an App Store & iTunes Gift Card in France, you can't redeem it in the App Store in the United States.

About Gift Card Scams -> About Gift Card Scams - Official Apple Support