On a character-twisted analogue of Sch\"{a}ffer'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 $n \ge 2$.