Hi Rorystew,
Welcome to the HP Forums, I hope you enjoy your experience! To help you get the most out of the HP Forums I would like to direct your attention to the HP Forums Guide First Time Here? Learn How to Post and More.
Sorry to hear that you are having issues with the printer not staying connected to your computer. I have seen this issue before and I was able to resolve the issue by simply plugging the power cable directly to a wall outlet. HP printers have a built in surge protector and if you have it connected to a power bar/strip, the printer will NOT get all the power it needs to function properly.
Here is a document that explains the Issues when Connected to an Uninterruptible Power Supply/Power Strip/Surge Protector.
Please NOTE that the printer listed in the document is a LaserJer printer and it is only used as an example. The document is meant for HP products in general.
Please feel free to write me back if you have any other questions.
Thank you for your time.