When we keep on stirring hot water vigorously it starts getting cooler. But we are increasing the random kinetic energy of the molecules of water. Heat is as it is the energy of RANDOM motion of atoms. So when we are stirring the water vigorously it should get hotter and hotter because the increase in energy of molecules of water. But the opposite happens..Why is this so? Has it something to do Mpemba effect
Getting cooler is the key phrase in you question. If you vigorously stir cold water you might get it to warm up if it was well insulated - duplicating Joule's clasic Mechanical equivalent of heat experiment. But if you stir a cup of hot coffee or a pot of hot water the far more significant effect effect it that the stirring accelerates cooling.
Stirring allows hot liquid to reach the surface much faster than unassisted convection. At the surface this hot water evaporates cooling the remaining water as a result.
Moving water convects heat better than static water. Take the reference frame of the water being static and the air moving. In this case, new cooler air is always sweeping in to take heat away. If the air is static, hot air remains at the interface and will not accept heat as well as cooler air. The same argument can be made for the moving water.
Edit: The motion you are providing by stirring in minimal compared to the microscopic motion of the water.