I am the American father of a child born out of wedlock outside the USA. Does my child have a claim to US citizenship?

Yes, if (1) you've been physically present in the US at least five years prior to the child's birth, two of which were after age 14. Honorable US military service, employment with the US Government or intergovernmental international organization or as the dependent unmarried son and member of the household of a parent in such service or employment, may be included AND (2) a blood relationship has been established between you the father and the child, and you agree to support the child until 18 years, and while the child is under 18 (i) the child is legitimated, (ii) you acknowledge paternity, or (iii) paternity is established by court adjudication.