Wow, it's great that you have his letters.
The UK National Archives at Kew have a partial muster roll for the Hompesch hussars from 1799 which may contain details on your ancestor. The reference for the record is WO 12/11742:http://yourarchives.nationalarchives.gov.uk/index.php?title=...
The next time I'm in the London area I hope to visit the archives and will try to check the Hessian records.
I believe there are also records at the German state archives concerning the Hessians but I have not looked into this in detail yet.
The best of luck with your research,