The hot water is not mixed with the cold water exit the discharge check valve. The hot water is only mixed with cold water at the faucets. As the hot water leaves the hot water tank, it is replaced with cold water at the bottom of the tank. The hot water heater in your home has no check valve in the hot water discharge line. The system works on pressure drop, i.e., as the pressure drops on the outlet of the heater due to flow, the cold water is ushered in by pressure form the cold water line. So I have been told, and it makes sense to me, that the only purpose for the check valve in the hot water discharge line is to prevent from pumping antifreeze into the hot water heater tank during the winterizing operation. Of course, I stand to be corrected about the check valve's function. If some out there has a better source of information concerning the use of a discharge check valve, I am all ears.
Before I would do anything else, I would do as suggested and check the condition of the discharge check valve, BTY - I replaced my plastic check valve with a brass check valve.