Closed-Form Expression for a Certain Sum Over Roots of Unity

Question

Let $p$ be an odd integer $\geq3$, let $n$ be an arbitrary positive integer, let $c$ be a non-zero complex number, and let $\omega$ be a complex variable. Does anyone know of a closed-form expression for the sum: $$\frac{1}{p}\sum_{k=0}^{p-1}\frac{e^{-\frac{2k\pi i}{p}\left(\frac{p+1}{2}\right)}}{\left(\omega e^{\frac{2k\pi i}{p}}-c\right)^{n}}=?$$