Power values of sums of certain products of consecutive integers and related results

Let $n$ be a non-negative integer and put $p_n(x)={\prod }_{i=0}^n(x+i)$. In the first part of the paper, for given $n$, we study the existence of integer solutions of the Diophantine equation $$y^m=p_n(x)+\sum _{i=1}^k{p}_{a_i}(x),$$ where $m\in {\mathbb{N}}_{\ge 2}$ and $a_1<a_2<\dots <a_k<n$. This equation can be considered as a generalization of the Erd\H{o}s-Selfridge Diophantine equation $y^m=p_n(x)$. We present some general finiteness results concerning the integer solutions of the above equation. In particular, if $n\ge 2$ with $a_1\ge 2$, then our equation has only finitely many solutions in integers. In the second part of the paper we study the equation $$y^m=\sum _{i=1}^k{p}_{a_i}(x_i),$$ for $m=2,3$, which can be seen as an additive version of the equation considered by Erd\H{o}s and Graham. In particular, we prove that if $m=2,a_1=1$ or $m=3,a_2=2$, then for each $k-1$ tuple of positive integers $(a_{2},\ldots, a_{k})$ there are infinitely many solutions in integers.