The Hessians where not mercenaries.
The British crown paid Hesse-Kassel to deploy part if its armies in support of the British.
That was completely normal in the 18th century.
At the Battle of the Boyne in Ireland in 1690 a good part of the danish army was involved... It was simply rented by William III.
At Blenheim in 1704 we again see part of the danish army... again rented.
One of the things that helped Brandenburg (Prussia) become a regional power was that it had a very good army. So other states paid them large sums of money for fighting on their side in wars. For some of the smaller German states renting out its army became part of how the state generated a income.
And the British paid Hesse-Kassel money for simply having a big army... so they could rent it if needed.
The men where still enlisted soldiers in the army of Hesse-Kassel.
It should be noted that during this period it was not in any way uncommon to be serving in a army from another state than where you where born.
During the Napoleonic wars the brits paid most of their allies money to help them keep larger armies in the field than they would otherwise have been able to.
And this is still being done... with UN missions the most comparable example.
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During the civil war there where attempts at recruiting men in Europe. The idea was to get men who where thinking about immigrating anyway. The union pay the trip and a big enlistment bonus. So if you survive the war, you will be way better off then if you had to pay your own way.
This is closer to mercenaries. But they are still enlisting at the same basic conditions as any american. And in the same units.
Obviously enlisting none citizens is done today also...
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edit: typos