Final answer:
When encountering a clergy member with a patient during a routine blood draw, the phlebotomist should greet both individuals respectfully and then ask the clergy member if they would like to stay or step out for privacy during the procedure.
Step-by-step explanation:
Upon entering a patient's room for a routine blood draw and noticing a clergy member with the patient, the phlebotomist should first greet both the patient and the clergy member respectfully. Then, the phlebotomist should briefly explain the purpose of the blood draw and ask if the clergy member would like to stay in the room or step out for privacy. It is important to respect the patient's wishes and provide a comfortable and confidential environment for the blood draw.