Yes, King Henry II of England did have an illegitimate son.
One of Henry II's notable illegitimate sons was Geoffrey, who was born around 1152. Geoffrey went on to have a significant ecclesiastical career. He was appointed as the bishop-elect of Lincoln and later served as the esteemed archbishop of York until his death in 1212. His lineage highlights the common practice of medieval monarchs acknowledging and providing for their natural children, often placing them in prominent positions within the Church or state.