How about Edward IV? According to Wikipedia -

Edward had numerous mistresses, the best known of whom was Elizabeth Shore, known as Jane Shore. He reportedly had several illegitimate children:

By Lady Eleanor Talbot: Edward de Wigmore (d. 1468). Reportedly died as an infant along with his mother.

By Elizabeth Lucy or Elizabeth Waite. Elizabeth Plantagenet. Born circa 1464, married Sir Thomas Lumley in 1477. Arthur Plantagenet, 1st Viscount Lisle (1460s/1470s – 3 March 1542).

By unknown mother: (Recent speculations suggests them as children by Lucy or Waite.) Grace Plantagenet. She is known to have been present at the funeral of her stepmother Elizabeth Woodville in 1492. Mary Plantagenet, married Henry Harman of Ellam, son of Thomas and Elizabeth Harman and widower of Agness. A daughter said to have been the first wife of John Tuchet, 6th Baron Audley.

Perkin Warbeck, an impostor claimant to the English throne, who claimed to be Edward's son Richard of Shrewsbury, reportedly resembled Edward. There is unconfirmed speculation that Warbeck could have been another of Edward's illegitimate sons.