The Diophantine equation $(x+1)^k+(x+2)^k+\cdots+(\ell x)^k=y^n$ revisited

Let $k,\ell\geq2$ be fixed integers, and $C$ be an effectively computable constant depending only on $k$ and $\ell$. In this paper, we prove that all solutions of the equation $(x+1)^{k}+(x+2)^{k}+\dots+(\ell x)^{k}=y^{n}$ in integers $x,y,n$ with $x,y\geq1$, $n\geq2$, $k\neq3$ and $\ell\equiv 1 \pmod 2$ satisfy $\max\{x,y,n\}＜C$. The case when $\ell$ is even has already been completed by the second author (see [24]).