Powerful numbers in the product of consecutive integer values of a polynomial

Let $n$ and $r$ be positive integers. Also let $k$ be an odd positive integer and $d$ be a non-negative integer. In this paper, we prove that if $k$ has at most four distinct prime factors, then the product $((d+1)^k+r^k)((d+2)^k+r^k)\cdots((d+n)^k+r^k)$ is not a powerful number for $n\geq\max\{r+d,59-r-d\}$. As a consequence, we prove that if $k$ has at most four distinct prime factors, then the product $(1^k+1)(2^k+1)\cdots(n^k+1)$ is not a powerful number.