Why coffee getting hotter after stir with milk?

Stirring a coffee involves a transfer of energy from you to the coffee. That’s because we are doing physical work on the coffee by pushing it around as it moves in the direction of your push. What began as chemical energy in your body becomes thermal energy in the coffee. That said, the amount of thermal energy you can transfer to the coffee with any reasonable amount of stirring is pretty small and you’d lose patience with the process long before you achieved any noticeable rise in the coffee temperature. The effect that you notice is more one of mixing than of heating. Until you mix the milk into the coffee, you may have hot and cold spots in your cup and you may notice that cold spots most strongly.