It sounds like you are powering your Roku from the USB port on your tvs, and that they are ports that power down when the tv is turned off (USB on some sets stays powered up).
I suspect what is happening is the tv is completing its startup faster than the Roku does, so when the tv checks for something connected to the HDMI port, the Roku is not yet ready to respond and the HDMI connection is never set up. Disconnecting/reconnecting the HDMI cable forces the tv to check again. You can likely force this recheck more easily by just switching to a different TV input and back.
The REAL solution is to power your Roku with the USB power adapter that came with it, plugged into your house power. The Roku will stay on and be ready the instant the tv powers up and checks the HDMI port. If you no longer have the USB power adapter that came with your device, you may have another one handy from another device that can be used -- just make sure the tiny print on it confirms it has a 5 volt dc output of at least 1 amp
Roku Community Streaming Expert
I am not a Roku employee, only a user like you. Please, no support questions via private message -- post them publicly to the Community where others may benefit as well.
If this post solves your problem please help others find this answer by clicking "Accept as Solution.".
Streaming Stick 4K+ 3821 | Streaming Stick+ 3810 | TCL Roku TV 43S245/C107X