Final answer:
The nurse should initiate a referral with a Speech-language pathologist.
Step-by-step explanation:
The nurse caring for a client with persistent problems with dysphagia, following a cerebrovascular accident (stroke), should initiate a referral with a Speech-language pathologist. Speech-language pathologists are experts in evaluating and treating communication and swallowing disorders, including dysphagia. They can evaluate the client's swallowing function and recommend appropriate interventions to improve swallowing safety and efficiency.