Using the substitution $u_i=x_i^p$ and letting $r:=q/p$ and $a:=1-1/p\in[0,1)$, we see that the integral in question is 
\begin{align}
	I&=np^{-n}\int_0^1\cdots\int_0^1\frac{du_1\cdots du_n}{u_1^a\cdots u_n^a}
	u_1^r\int_0^\infty dt\,e^{-t(u_1+\cdots+u_n)} \\ 
&=np^{-n}\int_0^\infty dt\,\int_0^1\frac{du_1}{u_1^{a-r}}\,e^{-tu_1}
\Big(\int_0^1\frac{du_2}{u_2^a}\,e^{-tu_2}\Big)^{n-1} \\ 	
&=np^{-n}\int_0^\infty \frac{dt}{t^{(n-1)(1-a)}}\gamma(1+r-a,t)\,
\gamma(1-a,t)^{n-1}, 	
\end{align}
where $\gamma$ is the [lower incomplete gamma function][1]. 

[1]: https://en.wikipedia.org/wiki/Incomplete_gamma_function#Definition