Final answer:
Increased lethargy is the additional finding that most indicates a potential deterioration in a client's neurological status following a closed head injury and should be immediately reported to a healthcare provider when accompanied by vomiting.
Step-by-step explanation:
When a client with a closed head injury begins to vomit, the nurse should report additional findings that may suggest a worsening condition. Among the options provided, increased lethargy, when combined with emesis, is a significant finding that the nurse should report to the healthcare provider. Increased lethargy could indicate that the client's neurological status is deteriorating, which requires immediate attention. In contrast, a heart rate of 80 beats/min is within normal limits, a sodium level of 145 mEq/L is also normal, and facial symmetry would not typically be associated with a new or worsening neurological issue in this scenario.