Final answer:
During thoracic surgery, specifically while removing a mass from the superior mediastinum, surgeons must take care to avoid damaging the left vagus nerve, which courses posterior to the arch of the aorta.
Step-by-step explanation:
The important nerve that the surgeon must be careful of while removing a thymic tumor from the superior mediastinum and that lies on and partly curving posteriorly around the arch of the aorta is the left vagus nerve (option C).
This is because the left vagus nerve is situated in close proximity to the arch of the aorta and is therefore at risk during procedures in this area, unlike the phrenic nerve, which innervates the diaphragm and lies more ventrally (towards the front) relative to the aorta.