Hello mpbeve, and welcome to the HP Forums, I hope you enjoy your experience!
I understand you're having problems with your ink quality. I would like to try and help you with that!
I would recommend going through the steps included in this document on Copy Quality Issues. There are many helpful steps that should resolve this issue.
You expressed concern that "one of the cartridges that has been installed for a long time is malfunctioning and requires replacement." Depending on the age of the cartridges, it could also be a matter of your ink needing to be replaced. This document on Limited Warranty for HP Ink Cartridges will direct you on the end warranty date of the cartridges themselves, which in turn will help determine if the cartridges need to be replaced.
Good luck and please let me know the results of your troubleshooting steps. Thank you for posting on the HP Forums!