You child automatically derives citizenship from you. It is that simple. You need to apply for Certificate of Citizenshhip for your child. Attach fee, photos, your naturalization certficate, his birth certificate.
Now issue of green card: you never applied for green card for him? If no, then what is his current status? Did he make illegal entry into US?
I am assuming he came with his mother on immigrant visa, later when eligible mother applied and obtained green card. However, they did not seem necessary for the child to get get green card. If that is the case, you need to also attach documents that prove his entry was legal. Which is why I am suggesting you attach his passport copy (with pages showing entry stamp). It should read, A#.... and Asylum granted.
Does the child have his original (from country of origin) passport?