Final answer:
The reddening of the eyes and the noted infection likely refer to the conjunctiva, which when inflamed, is known as conjunctivitis.
Step-by-step explanation:
When a nurse observes that a client's eyes are reddened and the client mentions an eye infection, it is most likely that they are referring to an infection of the conjunctiva. The conjunctiva is known to be a frequent site of infection and inflammation of this area is known as conjunctivitis, often resulting in a pink appearance of the eye. This is different from infections of the cornea (keratitis) or the sclera, as they are deeper structures of the eye that are less commonly infected. The client's description suggests that they are experiencing conjunctivitis.