The energy controller state warning is something you tend to see a lot with bluetooth adapters and can probably be ignored in this instance. I suspect that the authentication error in the log represents your real problem to resolve.
First, batteries. How long has it been since you've changed the batteries in your K380? If the keyboard isn't going to low power mode (related to the power controller warning), it is going to go through batteries much faster and they aren't going to last for 24 months like Logitech claims. If you haven't recently, try swapping them out for some fresh ones. At the time you do it, make sure to look over the battery compartment for any signs of corrosion (the K-series keyboard I previously had eventually died of this. I also remember that it was also finicky about bluetooth adapters it would work with as well).
If replacing batteries doesn't work, have you tried to remove the bluetooth software (including the drivers) and reinstall? If not, uninstall the bluetooth adapter software, reboot the system, then reinstall. To totally remove the actual bluetooth drivers (which often times are downloaded by Windows in the background when you plug in the BT adapter and remain installed after you unplug the adapter), you may have to go into Control Panel, hit the View menu, select the "show hidden devices" option, then manually right-click on each BT driver instance and remove it.
If neither of these works to improve the situation, you may want to pick up a higher quality bluetooth adapter of a different brand to try.