It took forever, but I was finally able to get the Amazon Prime Photos iPhone app to fully initialize and allow me to upload a file.
In my case the file ended up in Amazon Drive under Pictures/[iphone name]. An upload via the Amazon Prime Photos web client ended up in Pictures/Web
If you click on a picture in the Amazon Prime Photos web client and then click on the "i" (information) icon at the top, it will show you the file name. Copy that name and search for it in the Amazon Drive web client. It should find it and show you where it is.