There exist no closed-form expressions for arbitrary $d$ for the integral over the unitary group $\mathbb{U}(d)$ of a rational function of the matrix elements.     
There are asymptotic results for large $d$, see for example <A HREF="https://arxiv.org/abs/cond-mat/9604059">J. Math. Phys. 37, 4904 (1996)</A>. The leading order term is
$$\int_{\mathbb{U}(d)}\dfrac{1}{\sum_{k,l}u_{ik}\overline{u_{il}}c_{kl}}dU\rightarrow \frac{d}{\sum_{k}c_{kk}},$$
independent of the index $i$.

Alternatively, if $d$ is not large but the matric $C$ is close to the unit matrix, $c_{kl}=\delta_{kl}+\epsilon_{kl}$, one can expand
$$\int_{\mathbb{U}(d)}\dfrac{1}{\sum_{k,l}u_{ik}\overline{u_{il}}c_{kl}}dU=1-\int_{\mathbb{U}(d)}\sum_{k,l}u_{ik}\overline{u_{il}}\epsilon_{kl}dU+{\cal O}(\epsilon^2)$$
$$\qquad\qquad=1-\frac{1}{d}\sum_{k}\epsilon_{kk}+{\cal O}(\epsilon^2).$$