Assume that $\{p_1,p_2,\ldots,p_k\}$ are $k$ points on $\mathbb{C}P^1$. Let $\chi:\pi_1(\mathbb{C}P^1\setminus  \{p_1,p_2,\ldots,p_k\})\to \mathbb{C} \setminus\{0\}$ be  a group homomorphism.

> Is there a  singular holomorphic foliation by curve, briefly SHFC,  of  $\mathbb{C}P^2$  such that $\mathbb{C}P^1 \subset \mathbb{C}P^2$ would be a leaf of  the singular  foliation(actually the leaf at infinity)having $\{p_1,p_2,\ldots,p_k\}$ as singularities of the leaf at infinity and the linearized monodoromy representation  of $\mathbb{C}P^1$ coincide $\chi$? If the answer is no, for  what representation $\chi$, the  answer is affirmative?

By linearized monodoromy representation we mean the following:

We fix a base point $p$ on  the leaf at infinity $\mathbb{C}P^1$ which is not a singular point. We fix a $1$-dimensional local transversal $s$ passing $p$. For  every loop $\gamma$ based on $p$ and contained in $\mathbb{C}P^1$, we have a holonomy map $h_{\gamma}$  defined on $s$. Then $h_{\gamma}'(p)$ is  a  non zero complex number which is independent of homotopy class of $\gamma$  and is independent of choosing transversal $s$. So the  linearized monodoromy representation $\chi$  mentioned in the question  is defined as $\chi(\gamma)=h_{\gamma}'(p)$.