Is it permissible for my wife to pray in the children's room while my adult nephew is asleep in it?

A woman is obligated to perform prayer in the most secluded place away from eyes and the most covered place for her. The back rows are better for her, and her prayer in her home is superior, indeed, in the most hidden place within it, such as a private chamber. It has been narrated in a hadith: "A woman's prayer in her house is better than her prayer in her room, and her prayer in her private chamber is better than her prayer in her house." It is permissible for her to pray in a room with a mature non-mahram man, such as a sister's son, whether he is asleep or awake, provided she is fully covered with her hijab.