Welcome to HP Forums, this is a great place to get support, find answers and tips.
Thank you for posting your query, I'll be more than glad to help you out.
I understand that you are unable to print in black from your HP Photosmart 6510 e-All-in-One Printer.
Don't worry as I have a few steps that should help fix this issue.
This could be an issue with the printhead of the unit.
And here's the step by step guide for you in the link: Printer Prints a Blank Page or Does Not Print Black or Color Ink
Let me know how that works.
I hope you have a good day ahead.