How can I add a child’s phone to my account

I would recommend that your child get his or her own Apple ID. If you just log in with your Apple ID, then your child will also receive your messages, photos, etc.

Once your child has their own Apple ID, you can use Family Sharing. Then you have 100% control over which apps your child downloads, screen time, etc.

Here you can see how to set up Family Sharing

https://support.apple.com/kb/HT201088

If your child does not have an Apple ID:

https://support.apple.com/en-us/HT201084

I agree with the above.

To try to answer your question, use the article below as a guide to places where you need to sign into the child's phone to add the phone to your ID.

What to do after you change your Apple ID or password - Apple Support.