Let us consider the unit disc $\mathcal{D}=\{z: |z|\leq 1\}$ and an arbitrary continuous function $f:\partial \mathcal{D} \rightarrow \mathbb{C}$.
I want to find an extension of $f$ that is continuous on $\mathcal{D}$ and holomorphic on $int(\mathcal{D})$, by applying the Cauchy-Riemann integral formula: $f(z_0)=\int_{\partial \mathcal{D}}\frac{f(z)}{z-z_0}dz$.
Moreover, by Morera's theorem, a continuous, complex-valued function ƒ defined on an open set D in the complex plane that satisfies $\int_{\gamma}f(z)dz=0$ for every closed piecewise $\mathcal{C^1}$ curve $\gamma$ in $\mathcal{D}$ must be holomorphic on $\mathcal{D}$.
How can I apply the Morera's theorem to find out whether there is an extension of $f$ that is holomorphic on $int(\mathcal{D})$?
There is no such a thing as the “Cauchy-Riemann integral equation”; what you have in mind is Cauchy's integral formula.
See what happens if $f(z)=\frac1z$ when $|z|=1$. Then, if $0<|z_0|<1$,\begin{align}\int_{\partial D}\frac{f(z)}{z-z_0}\mathrm dz&=\int_{\partial D}\frac1{z_0}\left(\frac1{z-z_0}-\frac1z\right)\,\mathrm dz\\&=\frac{2\pi i}{z_0}\bigl(\operatorname{Ind}_{\partial D}(z_0)-\operatorname{Ind}_{\partial D}(0)\bigr)\\&=0,\end{align}and it is easy to see that $\int_{\partial D}\frac{f(z)}z\mathrm dz=0$ too. So, you get the null function in $\operatorname{Int}D$.