But I don't think they need to be declared to the ATP.
And they are not included in the ATP prize money because they are not part of the prize money winnings - they are like a commercial sponsorship, money for an ad campaign, a purely private arrangement.
My understanding is that quite a lot of events pay some appearance money. The Middle Eastern, Russian and Far Eastern tournaments were certainly known for it previously.
I can only imagine that when Andy plays tournaments like Shenzhen, he's guaranteed something likely well beyond the prize money available for going somewhere so out of the way.