Your post has been here a while, but since no one has stopped in to help you, then here I am!
I have been trying to do what you have requested to do, but I am afraid, you are right – there is no way to upload documents like Word , PDF and such to your My Cloud via the app…
Although, I have been able to upload a PDF to another of my online cloud servers, and also I could email the PDF to myself. So, all is not lost using the My Cloud app for uploads.
When I tapped on the upload icon at top of app screen, and I saw four choices, one was “Email”, and used that to send the file to myself. Another choice named “Open in” enabled me to see my other server (as you likely saw OneDrive) and I was able to send PDF to my other cloud server
Apparently, when using the app to access the My Cloud we cannot send something to My Cloud other than pictures,; we can only download from it. In fact, I recently triedto send a music file to My Cloud device and could not do it, as I recall. I will check into doing that again to see if I am correct about this.
Meanwhile, continue sending files to your OneDrive or email them to yourself.