I often put hot water from the tap into a pot when I want to boil something on my stove (although never when making tea!).
I think this is an efficient use of energy as I have already heated the water and stored it in the water heater tank. Moreover, I think that by adding a small amount of cold water to a large tank I will use less energy to bring the slightly cooled tank volume back to the original temperature (about 100 degrees F) than I would use to bring the same volume of cold water to boil on my gas fueled stove.
Given that abundant variables prevent calculating the actual energy of hot v. cold water, am I generally on the right track?