Final answer:
The nurse should first assess the client for airway patency when dealing with hemorrhaging from bleeding esophageal varices and an esophagogastric tube.
Step-by-step explanation:
The priority when caring for a client who is experiencing hemorrhaging from bleeding esophageal varices is to ensure airway patency. This is because an obstructed airway can rapidly lead to respiratory arrest and is immediately life-threatening. Therefore, the nurse should first assess the client for airway patency before proceeding with other interventions such as sedating the client, maintaining balloon pressure, or irrigating the gastric lumen.