#Note:

In this question by $\mathbb{C}P^1 \subset \mathbb{C}P^2$ we mean that we choose the line at infinity in the form $\{[0,y,z]\in \mathbb{C}P^2\} $ which is identified by $\mathbb{C}P^1$.

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 having $\{p_1,p_2,\ldots,p_k\}$ as singularities and the linearized monodoromy representation of $\mathbb{C}P^1$ coincide $\chi$? If the answer is no, for what kind of representation $\chi$, the answer would be 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 $\chi(\gamma)=h_{\gamma}'(p)$.