I want to compute
$$ \int_{0}^{2\pi}\frac{{\rm e}^{{\rm i}kt}}{\,\left\vert\,{{\rm e}^{{\rm i}t} - {\rm e}^{{\rm i}t_{0}}}\,\,\right\vert^{\, m}\,}\,{\rm d}t, $$
where $k\in\mathbb{Z}$, $t_{0}\in\mathbb{C}$, and $m = 1,3,5,\ldots$. The integrand experiences a singularity at $t=t_0$ and $t = \bar{t}_{0}$. We may assume that $\left\vert{\,{\rm e}^{{\rm i}t_{0}}\,}\right\vert < 1$. However, if someone is able to make any progress for any $m$ and e.g. for only $k > 0$, it would be of much help.
You may also consider the integral rewritten into a contour integral over the unit circle $S$ by letting $\xi\left(t\right)={\rm e}^{{\rm i}t}$ and $\mu = {\rm e}^{{\rm i}t_0}$,
$$ \frac{1}{i}\oint_{S}\frac{\xi^{k - 1}}{\left\vert\xi-\mu\right\vert^{m}}\,{\rm d}\xi. $$
Another reformulation is to let $t_{0} = a_{0}+ {\rm i}b_{0}$ and $\alpha = {\rm e}^{-b_{0}}$. Then, the integral can be found as
$$\int_{0}^{2\pi} \frac{\cos\left(kt\right) + {\rm i}\sin\left(kt\right)} {\,\left\{\left(1 + \alpha\right)^{2} - 2\alpha\left[1 - \cos\left(t - a_{0}\right)\right]\right\}^{m/2}\,}~\text{d}t,$$
where one may treat the two integrals separately.
I have not been able to make any significant progress on any of these three formulations.