I'll add that I think if you want them ALL gone, they can likely do it easily. But if you want to keep any, that's where the problem comes.
The Export Notes and Import Notes functions could help with that selective case . Once you see how the notes are formatted, you can write a utility to create an edited file, replacing with nulls the data that you don't want, but keeping the horse's name, or race number, etc so it writes over what's currently in your notes file on their server.