In short,
$$ \begin{cases}
\text{when }1\leq x & \text{series diverges when }y\le1\\
\text{when }\frac{1}{2}<x<1 & \text{series diverges when }y\leq\frac{x}{2x-1}\\
\text{when }0<x\leq\frac{1}{2} & \text{series always diverges.}
\end{cases}
$$

When $x>1$, the inner sum of $$\sum_{k=1}^{\infty}\frac{1}{k^{y}}\sum_{h^{x}\leq k^{y}}\frac{1}{h^{x}}$$
converges so this series diverges precisely when $y\leq 1$. When $x=1$, since $H(k^{y})\sim y\log k$, again we see that this diverges for $y\leq 1$. Lastly, when $0<x<1$
we have that $$\sum_{h^{x}\leq T}\frac{1}{h^{x}}=\int_{1}^{T^{1/x}}\frac{1}{s^{x}}d\left[s\right]\sim\int_{1}^{T^{1/x}}\frac{1}{s^{x}}ds\sim\frac{1}{1-x}\left(T^{1/x}\right)^{1-x}=\frac{1}{1-x}T^{1/x-1},$$
and so the convergence of the series depends on the convergence of $$\sum_{k=1}^{\infty}\frac{1}{k^{y}}k^{y\left(\frac{1}{x}-1\right)}=\sum_{k=1}^{\infty}k^{y/x-2y},$$
and this depends on when $\left(\frac{1}{x}-2\right)y<-1.$
Thus it diverges for every value of $y$ when $0<x\leq\frac{1}{2}$ , and for $\frac{1}{2}<x<1$
, it diverges when $y\leq\frac{x}{2x-1}.$