When you sign out of your ID on the old phone it will leave any content on that phone unless you erase it. However, without your ID the child will not be able to update the apps, or redownload them, without your account information.
If you do it by leaving things behind, make sure you delete any private information. You may have to do that individually in each setting that may have it. Normally if you are giving a phone to another user:
What to do before you sell, give away, or trade in your iPhone, iPad, or iPod touch - Apple Support
If you do decide to do it "by the book" with fully erasing, you can set up Family SHaring so you can share the apps. However, if the child has been using your device to pay these games I suspect things like scores may be lost if the app stores them on the device instead of online..
Family Sharing - https://support.apple.com/HT201060
Set up Family Sharing - https://support.apple.com/HT201088