"This child is my relative."
Translation:Αυτό το παιδί είναι συγγενής μου.
If an article was used, then it would be implied that there were no other relatives. Contrast with "Αυτός ο άνδρας είναι ο οδηγός μου" - This man is my driver : It is implied that there are no other drivers. To summarise: If it is implied there is only one (e.g. my father) an article is used, otherwise, none.
The concept is a bit similar to "My friend came" Vs "A friend of mine came".
You are right about "συγγενής" being both a noun and an adjective but with different meanings. As a noun it refers to a member of your family. In this case it can only be masculine (ο συγγενής in singular / οι συγγενείς in plural) or feminine (η συγγενής in singular / οι συγγενείς in plural). As an adjective (ο συγγενής, η συγγενής, το συγγενές in singular / οι συγγενείς, οι συγγενείς, τα συγγενή in plural) it means something that has common origins or common traits with something else. E.g. συγγενείς γλώσσες (languages that share common origins or common traits) "Συγγενής" as an adjective in medical context can also refer to a congenital disease.