PythonAnywhere dev here -- you don't need a run.py on PythonAnywhere. The code that normally goes in there is to run a local Flask server that can serve your app -- that's all handled for you by our system.
Instead, you need to change the WSGI file (linked from the "Web" tab) to import the appropriate application module. So, because the sample site you have on github does
from app import app
...on PythonAnywhere in the WSGI file you'll need to do this:
from app import app as application
One thing to be aware of -- if I'm understanding your file listings above correctly, you don't have all of the github app installed -- only the templates. You'll need __init__.py, hamlet.py, and config.py, and they'll need to be in the same directory structure as the original.
Thank you, changing the WSGI file and keeping the original directory structure worked!!! Kudos to pythonanywhere, it made web deployment much easier than it was !!!