The troops in the photo are members of the 3rd company. The 3rd company was the only unit in IR 138 to be awarded the Kaiser Prize. The 3rd company won the Kaiser Prize in 1912.
Here is a photo of the Infantry Kaiser Prize. The prizes were awarded from 1895 through 1913.
The only complete listing of army units (infantry, jaeger, machinegun, field artillery & foot artillery) that won the Kaiser/Kings prize of which I am aware was published in a German language collectors publication in the 1970s. If you are interested in a specific unit, semd me a PM and I will be happy to look up the unit.