Adding milk first increases the volume to heat and lowers average temerature, but adding it afterwards seems to have similar effects. How can you compare the two?
0: get a watch, a thermometer, two equal doses of milk and two of coffee and a microwave. As you have a computer to make the annotations, a pen and a stencil are not needed.
1 : write down a proper definition of 'efficiency', choose one at your will.
2 : do the proposed first experiment, time the results and the temps. Taste ;-)
3 : do the proposed second experiment reversing the order, time the results and the temps. Taste ;-)
4 : compare the resultant temps and tastes (is a bit subjective but it is important) with the settled goal (efficiency deffinition).
5 : kindly let us know your conclusion.
You may think that I am making fun, but it is not so. The experimental way, when available, is the best solution. You have a courious mind, dont stop. A personnal learning process is good.
To be more rigourous one can invent a physical model (between point 1 and 2) to advance a candidate outcome. But the point 1 is allways indispensable.
end of recipe ;-)
By adding milk first you reduce heat loss during heating due to lower average temperature. The rate of heat transfer to the surroundings depends on the difference in temperature between the object and its surroundings, so lower average temperature-> less energy lost to the surroundings.
So if you add milk first, you will get higher temperature at the end.
But personally, I prefer to add milk after, as it's more sensetive to overheating. You can heat cofee to 100C without any damage to it.
A very similar experiment, of testing whether tea or milk was first added to the cup, was used to demonstrate the scienfific methods of hypothesis testing by Ronald Fisher (see Lady tasting tea). The art of hypothesis testing has been largely forgotten by mainstream physicists, due to the high precision of physics measurements, but it continues to be at the core of scientific research elsewhere - e.g., in biology, medicine, etc.
Obviousy, the experiment does not answer why the taste is different, it only confirms that the difference exists.