HP printer drivers

As you probably have discovered, Mac OS X 10.4 appears to be the earliest supported version.

Printer Specifications for HP Photosmart Premium (C309a and C309c) All-in-One Printers | HP® Customer Support

HP Photosmart Premium Fax All-in-One Printer - C309a Software and Drivers | HP® Customer Support

A workaround could possibly be to produce PDF files (using a commercial Acrobat program or the free PrintToPDF printer driver) on a Mac OS 9 computer, and then transfer these files to a supported Mac (or PC) for printing. For use in a local network, one may want to install a small FTP server (such as the now free NetPresenz) on the Mac OS 9 machine, allowing direct download/upload to/from an FTP client on almost any modern computer.

http://www.jwwalker.com/pages/pdf.html

http://www.stairways.com/main/netpresenz

do you have a newer computer running OS X that can print to that printer with the appropriate drivers, and Share it so that the OS 9 computer can print to the Shared printer?