An illegitimate child or bastard was one that had been conceived out of wedlock, where the latter term could also be used as a more general slur against the character of the individual referred to thusly. In both cases, the child in question was often seen as an embarrassment.
In 2285, when Kruge had Kirk's son David Marcus killed, Kirk angrily called him a "Klingon bastard!" (Star Trek III: The Search for Spock)
In 2365, Debin accused Thadiun Okona of fathering his daughter Yanar's "bastard child." In reality, the baby was the child of Yanar and Benzan. (TNG: "The Outrageous Okona")
In 2367, Toral, son of Duras' parentage was in question, as Duras had no known mate. When B'Etor suggested that a genetic scan would prove the Klingon's bloodline, Gowron protested the possibility of Toral becoming Chancellor anyway, saying "The illegitimate son of Duras cannot rule the Klingon High Council." (TNG: "Redemption")