"He is not my cousin."
Translation:Non è mio cugino.
-The article is omitted when the possessive refers to a singular noun indicating kinship, such as: padre (father), fratello (brother), zio (uncle), cugino (cousin), suocero (father-in-law) etc...- (with some exceptions)
PS: "Fidanzato (fiance)" is not a family member, so the article is necessary - il mio fidanzato -
"mamma" and "papà" are exceptions, they are considered altered nouns, so the article is necessary