Which word describes a face that appears pale due to poor health?

The word that describes a face that appears pale due to poor health is "pallid." This term specifically refers to a deficiency of color, often associated with illness or fatigue. When someone is pallid, it suggests they may be unwell or lacking vitality, which can be visibly observed in their complexion.

The other words, while having distinct meanings, do not relate to the physical appearance of a face in the context of health. "Prolific" refers to producing abundantly, often used in the context of creative works or reproductive capacity. "Ebullient" indicates a cheerful or enthusiastic disposition, reflecting a lively personality rather than a physical state. "Flippant" describes an attitude that is not serious or respectful, which also does not convey any idea of physical appearance related to health. Consequently, "pallid" is the only word that appropriately fits the description in the question.