Let me show that such a map usually doesn't exist (even if we don't remove any additional curves). Consider the case when $\Gamma$ is a curve of degree $\ge 4$. Then one can slightly perturb $\Gamma$ in $\mathbb CP^2$ to a curve $\Gamma'$ that is not isomorphic to $\Gamma$ (so that it stays in a small neighbourhood of $\Gamma$). If the map $\pi$ would exists, we can restrict it to $\Gamma'$, and it will be extendable to the whole $\Gamma'$. So we get a holomorphic map $\pi: \Gamma'\to \Gamma$, which is impossible, since $\Gamma$ and $\Gamma'$ are not isomorphic (and by Riemann-Hurwitz for two curves of the same genus $\ge 2$ a non-constant holomorphic map $\Gamma\to \Gamma'$ exits only if the curves are isomorphic).

This argument can be easily generalised to curves $\Gamma$ of degree $3$, i.e. cubics. The case of conics and lines should be analysed  separately (and for lines $\Gamma$ such a map sometimes exists, for example, when all $\Gamma_i$'s are lines transverse to the line $\Gamma$ and intersecting at one point).