Well, if the channels are Clear QAM, do they play fine outside of Kodi? Tryin VLC, and from the File menu tell it to open a network stream, using
http://<HDHR IP>:5004/auto/v<ChannelNumber>@. If it plays fine that way, it's a either a Tvheadend issue or a Kodi issue.
Following the same idea of using VLC, this time select a service or channel in the Tvheadend web UI, and paste the link from Tvheadend into VLC's "Open Network stream" dialog. If it plays fine there, hen the problem isn't Tvheadend, it's Kodi.
Another thing to check: make sure that Tvheadend is using the correct frequencies. It could be cable, or hrc, or irc. To see what your device says, from the command line, run @hdhomerun_config <DeviceID> get /tuner0/channelmap@. This will display which mapping your tuner is defaulting to. To see if the frequencies match, go to @http://<HDHR IP>/lineup.html
and select a channel to play. (If it doesn't open in VLC, copy the play link into VLC's "Open Network stream", as before) and let it play. Then in a command prompt, run
hdhomerun_config <DeviceID> get /tuner0/channel
and @hdhomerun_config <DeviceID> get /tuner0/program@. The channel and program should match the mux and service in Tvheadend. If they don't, then there's a problem with your Tvheadend configuration and you're using the wrong US cable frequencies.
Those options ought to get you going and be able to narrow down the problem.
