Let $D$ be an effective $ \mathbb{Z}$divisor on $ \mathbb{P}^1$. Is there a form to associate a curve $C$, and morphism $C \to P^1$ to the divisor $D$ ? For example, let $Y$ be a singular plane curve, sometimes, we can built a cover of $ \mathbb{P}^2$ that branches along $Y$. Is there a similar construction for covers of $\mathbb{P}^1$ ? I will appreciate any reference in this direction, or anything related? Thanks
Yes, this is the famous Riemann Existence Theorem. In its general form, it can be stated as follows. Theorem. Let $Y$ be a compact Riemann surface, and $D \subset Y$ be an effective reduced divisor. Then there is a $1$$1$ correspondence between the following sets: $\mathbf{1)}$ finite covers $f \colon X \to Y$ of degree $d$ whose branch locus lies in $D$, up to isomorphism; $\mathbf{2)}$ group homomorphism $\rho \colon \pi_1(Y  D) \to S_d$ with transitive image, up to conjugacy in $S_d$. Now, the fundamental group of $\mathbb{P}^1 \{b_1, \ldots, b_k\}$ is the group generated by $k$ generators $\gamma_1, \cdots, \gamma_k$ with the unique relation $\gamma_1 \gamma_2 \cdots \gamma_k=1$, where each $\gamma_i$ is the homotopy class of a small loop on $\mathbb{P}^1$ around $b_i$. Then we obtain the following corollary: Corollary. Let $D=\{b_1, \ldots, b_k\} \subset \mathbb{P}^1$ be a finite set of points.Then there is a $1$$1$ correspondence between the following sets: $\mathbf{1)}$ finite covers $f \colon X \to \mathbb{P}^1$ of degree $d$ whose branch locus lies in $D$, up to isomorphism; $\mathbf{2)}$ conjugacy classes of $k$tuples $(\sigma_1, \ldots, \sigma_k)$ of permutations in $S_d$ such that $\sigma_1 \sigma_2 \cdots \sigma_k=1$ and the subgroup generated by the $\sigma_i$ is transitive. Notice that neither $X$ nor the degree $d$ of the cover are uniquely determined by $D$. For more details, see Miranda's book [Algebraic curves and Riemann surfaces, p. 9092]. 

