Final answer:
The most appropriate next step in the management of this patient is platelet transfusion.
Step-by-step explanation:
The most appropriate next step in the management of this patient is platelet transfusion. The patient's presentation is consistent with thrombocytopenia, which is a condition characterized by low platelet count.
This can lead to abnormal bleeding, such as the recurrent nosebleeds described by the patient. Platelet transfusion can help restore the platelet count and prevent further bleeding.