Is my request to the master's student I supervise, to buy an item for me from his travels, with payment of the price and expenses, considered an exploitation of my authority, and does paying an additional fee remove this suspicion?

There is no harm in you asking the student to buy something for you during his travel, as long as you do not compel him to do so, and you pay the price and the costs of bringing it. It is preferable that you give him some money in return for his favor, even though you are not obliged to do so.