Let $B = \{x_1,\dots,x_{d-2},y_1,\dots,y_k\}$ be a subscheme of $d-2+k$ distinct points of $\mathbb{P}^1$, and $g:B\rightarrow \mathbb{P}^2$ be a morphism mapping $x_1,\dots,x_{d-2}$ to a fixed point $p_0$ and $y_1,\dots,y_k$ to general points $q_1,\dots,q_k\in\mathbb{P}^2$.

Assume that there exists a morphism $f:\mathbb{P}\rightarrow\mathbb{P}^2$ of degree $d$ restricting to $g$ on $B$, so that the space $Hom_d(\mathbb{P}^1,\mathbb{P}^2,f_{|B} = g)$, of morphisms restricting to $g$ on $B$, is non empty.

Do we have that $h^1(\mathbb{P}^1,f^{*}T_{\mathbb{P}^2}\otimes\mathcal{I}_B) = 0$ for $d\gg 0$? Would this imply that $Hom_d(\mathbb{P}^1,\mathbb{P}^2,f_{|B} = g)$ has dimension $h^0(\mathbb{P}^1,f^{*}T_{\mathbb{P}^2}\otimes\mathcal{I}_B) = d-2k+6$ and that it is smooth at $f$?

Thank you very much.