Hi!
Welcome to the HP Support Forums. I understand you are getting an 'Error: 0x610000e' on your Photosmart C309a printer.
I have included a trouble shooting document with detailed steps to take when you receive an alphanumeric code on the front panel of your printer.
An Alphanumeric Code (Example: 'Ox…' or 'C2…') or 'Ink System Failure' Error Message Displays on the Front Panel When Printing
http://h10025.www1.hp.com/ewfrf/wc/document?cc=us&lc=en&docname=c03081973
Thank you for the updated information on your issue.
In your first post you indicated that you had a network issue as well. I wonder if while correcting the network issue if the printer lost its connection to your wireless network. Please try running the Wireless Setup Wizard from the front panel of your printer to ensure the printer is connected to your wireless network. I have included your user guide, which details how to run the wireless setup wizard. The steps can be found on page 7 under the section titled "Use Wireless Setup Wizard".
HP Photosmart Premium Fax C309 Series User Guide
http://h10032.www1.hp.com/ctg/Manual/c01659847.pdf