I have a 4050 with a JetDirect network card, and it mostly* works under Mavericks.
HP's web page shows drivers for the Mac only for up to OS 10.7, and support for later OS versions is not likely to be forthcoming. Instead, the latest drivers for all attached printers are automatically downloaded and installed by Software Update - but only for printers connected and communicating with the OS. You might want to connect the printer via USB when you run Software update, to ensure that the printer is noticed. You'll need a USB-parallel adapter (~$15) to do it.
It does sound, however, like you have a bad network card in the printer. If so, the USB adapter ought to (sort of) solve the problem. A used nework card is about $25 on eBay.
*Apple's drivers are not free of bugs: ever since the Mavericks upgrade, specifying "landscape" fails to rotate the text in text boxes. (The boxes rotate, but not their contents!) I had to re-do some envelope, label, and check templates.
Thanks for the continuing support on this problem. It was working fine, then it suddenly stopped working. I fixed it once by manually reentering the IP Address in System Preferences. Then it stopped working again. Reentering the IP address didn't fix it this time.
I doubt that it's the network card on the HP because it was working flawlessly until I updated to Mavericks (grrrr).
I will try the USB-Parallel adapter next.