Try this: "A mercenary, sometimes known as a soldier of fortune, is a private individual, particularly a soldier, who takes part in military conflict for personal profit, is otherwise an outsider to the conflict, and is not a member of any other official military. Mercenaries fight for money or other forms of payment rather than for political interests."
These guys were professionals serving in their principality's army, in recognized units of that army, commanded by officers in that army. They were sent to NA to serve with the British Army under the ultimate command of Howe/Burgoyne/Clinton and did so. Whether they were sent there because the Landgrave believed in the Crown's cause, because he was afraid that a successful rebellion in NA might cross the Atlantic to central Europe, because he was related to the House of Hanover, or because he saw a way to fatten the Hesse-Cassel public fisc, is irrelevant to the fact that they were performing their military duty. And you seem confused about why they "got pay". They "got pay" for following the orders of their C-in-C. You concede this at one point: "They fought because their Prince or King took a bag of gold coins from King George III and told them to march off to a war". Yep. It's called "following orders", which is what a professional soldier does. That's not what a "mercenary" does.