"It is like Hughes has blocked the Amazon Prime streaming app for some reason."
That's definitely not it, as HughesNet does not block Prime or any other streaming services. It's also likely not on HughesNet's end but on the internet side, and possibly Amazon's service not working well with latency and congestion, or with your Fire stick not playing nicely with latency and congestion, or with your smart TV.
Do you have a laptop or desktop? Try watching Prime that way and see if it works.
I don't have a Fire TV Stick, so I don't know what their disconnection notice actually means, but if you can view Netflix through it without issue, but not Amazon Prime, it sounds more like a congestion and/or latency issue than anything going on with the equipment. If there were an actual connection issue, such as with the WiFi or with the Fire TV Stick connecting to the net at all, it's likely that it would be doing the same for anything you tried to use it for, including Netflix.
Amazon Prime is known to be the most "touchy" when it comes to streaming with HughesNet, unfortunately.
This is only a guess, however.
That's assuredly congestion, or a combination of congestion and latency.
With this said, you can run a few speed tests to ensure that your speed is okay, but even with adequate overall speed the congestion can still be there, causing a problem. To run the speed tests, please see these instructions.
As for your phone, that's not good. If you don't already have it, I would download the HughesNet Mobile App and run its WiFi signal test. This will tell you how good of a signal it's getting, which can not only be affected by range, but also by signal interference from other devices. I would especially test it when you experience a disconnect. You can get the HughesNet Mobile App at the App Store and Google Play for Apple and Android devices, respectively.