Can produce flame do damage on unarmed strike?

So I have seen previous questions regarding this situation, but all of them appeared to be for fun and silly one shots. What’s different in this case is I’m trying to run a long and intense campaign. The monk of the party recently gained the feat magic initiate and took the Druid cantrip produce flame. He asked me about when he used an unarmed strike while his hand was on fire if the damage from the fire would be added. I wasn’t sure so I just added a small amount of fire damage at the time. After doing f some research I understand that the flaming hand can do other things like holding a sword or a shield, but I don’t know if I should allow him to add damage. By the way I am new to DnD so I’m sorry if I missed something obvious which would answer my question.

Thank you for your time.