On a character-twisted analogue of Schäffer's equation

Let $f$ be a positive integer, and let $\chi$ be a primitive quadratic character of conductor $f$. Let $k$ be a positive integer, and write $B_k(\chi ,X)$ for the $k$-th Bernoulli polynomial corresponding to $\chi$. Suppose $B_k(\chi ,X)$ is irreducible and of degree at least $2$. Then for 100% of positive integers $m$ divisible by $f$, the Diophantine equation $$\chi (1)\cdot (x+1{)}^k+\chi (2)\cdot (x+2{)}^k+\cdots +\chi (m)\cdot (x+m{)}^k=y^n,$$ has no solutions with $x$, $y$, $n$ integers, and $n\ge 2$.