Let $k⩾2$ be an integer. We studied the number of integers which form perfect $k$-th powers in the way of

$$x(y_1^2+y_2^2+y_3^2+y_4^2)=z^k.$$

For $k⩾4$, we established a unified asymptotic formula with a power-saving error term for the number of such integers of bounded size under Lindelöf hypothesis, and we also gave an unconditional result for $k=2$.