For $y=1$ it is the average of $0,1,...,x-1$ giving $(x-1)/2$. This is obvious but is the first case, which can be written as
For $y=2$ it is
which comes to
For even as small as $y=3$ the similar iterated sum didn't evaluate to anything nice when put into maple; the result involved the $\Psi$ function and the constant $\gamma.$
But the summation (not closed form) version for $y=3$ is obtained by stringing along one more variable:
$$(1/x)\sum (1/x_2) \sum (1/x_3) \sum x_4,$$ where each sum goes from 1 to one less than the next outer variable.
Given the intricacy of maple's closed form for the $y=3$ case, I would be surprised by a closed form for the case of general $y$.