Seems that this app requires internet connection to work. And hp do state in their "getting started" section that it won't work with a straight usb only connection...
Overall it appears that they use a web-based server system to allow the app to print...? (Likely explaining that rather poorly; more a guess on my part as don't entirely follow how they have designed and configured the app overall - not one I'm familiar with.) Seems like a path similar to M$ Office 360 (which is a cloud based Office suite?)
I have tried a few printer apps for use with my iPad2 - and while they support some printers (enables/allows them to print directly from the iPad - via wifi of course) there are many it won't. (My elderly Epson laser isn't supported (it's circa 1995...); likewise my Canon mp6003 - early 2000's?) Those not supported for direct printing must then use a computer as the print server, and the app/iPad sends the data to the printer that way. That computer of course has to be powered up etc...
None of the apps I have tested require an internet connection to allow printing; and the print action is there when you select the printer required.
There are quite a few more recent printers that are compatible with many of these apps; will allow such apps to print direct via/to the printer - no "server" of any kind involved.
Perhaps checkout a few more apps; and many allow a try before you buy option - limited functionality but a try out at least.