Requesting a refund for an Apple Pay transaction sent to the wrong number

I believe you specifically mean Apple Cash sent through Messages. Those are cash transactions like any other where you would need to ask for the money back from the person you sent it to, if they have already accepted it. No different then if you gave someone on the street $20, your bank is not going to refund your money if that person does not give you the money back.

Apple Cash person-to-person payments are like any other private transaction between two people, so only send and receive money with people you know.

Send and receive money with Apple Cash - Apple Support

In most cases the person you would know would be one that has their number stored in your Contacts and not one where you would be manually entering a number.

Apple Pay is basically a cash transaction. Apple will not refund money you sent to the wrong number. But you may have a couple options other than contacting that wrong number and asking them to send the money back. Please see the discussion linked below.

How do I request a refund for an Apple Pa… - Apple Community

You used Apple Cash not Apple Pay. Apple Cash is just like it sounds, cash. You made a mistake. I understand, but neither Apple or your bank are responsible for your mistakes.

If the person you sent it to won’t refund the transfer, you’re not getting it back. sorry, but it’s just like handing cash to a stranger on the street. You’re unlikely to get it back.