We first find the $ x $, the average of $ 13 $, $ -16 $, and $ 6 $ by summing the three integers and dividing the sum by $ 3 $, so we have that $ x = \frac{13+(-16)+6}{3}=\frac{3}{3}=1 $. Now, we try to find $ y $, the cube root of $ 8 $. We know that $ 2^3=8 $, so $ y=\sqrt[3]{8}=2 $. Finally, we can substitute in $ 1 $ for $ x $ and $ 2 $ for $ y $ to find $ x^2+y^3 $: $$x^2+y^3=(1)^2+(2)^3 = 1 + 8 = \boxed{9}.$$