Final answer:
The passage indicating Prince Henry's attempts to hide his fatigue best supports the inference that he is embarrassed about being tired, as covering up his state hints at discomfort with others knowing his true feelings.
Step-by-step explanation:
The detail that best supports the inference that Prince Henry is embarrassed to admit he feels tired is the passage that describes Prince Henry's attempts to hide his fatigue. This behavior implies a sense of embarrassment or a reluctance to show weakness by concealing his tiredness. The act of trying to hide something often aligns with feelings of discomfort or shame about what is being concealed. In contrast, merely yawning, apologizing, or feeling tired after a long day does not inherently indicate embarrassment. It's the effort to cover up his state that suggests he is not comfortable with others knowing how he truly feels.