Last April, I started a topic that my new HP Photosmart B109a-m printer/scanner wouldn't scan. Neither would the HP toolbox (Device Manager) work.
After much testing with various distros, I found that it would only work if I used HPLIP 3.9.12, not with later versions.
I have now found the answer, which I am posting here in case it is of any use to anyone.
It was an HPLIP fault with the later versions.
To solve it, here is what I did.
As root, open a text editor and go to /usr/share/hplip/data/models/models.dat
Find the line which reads [photosmart_b109a_series]
Change it to read [photosmart_b109a-m]
Save the file.
Scanning and the HPToolbox now work correctly, and the toolbox finds the printer and will set it up.
Hope this is of interest.