Printer stack on processing - failed to read ppd file
This warning means that the hp-check script doesn't have the permission to read this file.
You can simply change the permissions like this (open a terminal):
$ sudo chmod 644 /etc/cups/ppd/HP-LaserJet-Pro-MFP-m125nw.ppd
It will give read permission to everyone, and read/write to root.
I have an HP printer PhotoSmart and had the same problem... I went into /etc/cups/ppd/ and modified the permission of my ppd file (root permission needed): It was Owner Root Read and write - Group lp Read only - Other No right I changed that last permission into Read only And it did the trick so far! Good luck!