From the new biography on king Willem III by Dr. van der Meulen it appears that Willem III wanted to renounce his rights so he could settle on the French countryside with his pregnant mistress, the opera singer Louise Rouvroy. When his father dies in 1849 he decides to accept the throne after all. What happens to Rouvroy and their child is unknown, though she did work as singer in Paris for a while.
Willem I en III hadden bastaarden - NOS Nieuws
The rumours that the king could not have been the father of Queen Wilhelmina due him having syphillis is untrue. The king never received any medication for venereal diseases. And he was even present at her birth, which was very exceptional for those days. The biographer emphasizes that esp. the last 10 years, áfter his marriage with Emma of Waldeck-Pyrmont, the king became more stable and more productive even.