I am unable to connect my HS110 to my router. The plug just keeps blinking amber/green at the rate of once per second. Kasa simply tells me to connect manually but when I go into settings on my phone, Oneplus 3, I find that it reads connecte3d no internet and the wifi symbol on my phone is marked not connected and there is no wifi between phone and router.

TP-Link can offer no advice except to say that the plug and the router are not compatible, is this likely? And, anyone have any advice?

I'm not sure I follow your explanation, it sounds like what's expected. No internet will be available when you're connected to the HS110 during setup. The next step, after connecting your phone to the HS110, is to run through kasa to add a new device. If your phone is having trouble staying connected to the HS110 you need to disable any setting that's similar to "prefer wifi with internet connections" until you're finished the setup.

After the HS110 has been setup it will connect directly to your router, the HS110 signal will disappear and your phone will then connect to your router as well. At that point you should be able to control the HS110.