Asking for a deformation over $\mathbb{A}^1$ is quite restrictive. Even asking for formal deformations / deformations over an étale cover of $\mathbb{A}^1$ is nontrivial. The "standard" obstruction group for deforming a stable map is the hyper-Ext group $\mathbf{R}Hom^2_{\mathcal{O}_{C_0}}(L^\bullet_{f_0},\mathcal{O}_{C_0})$, where $L^\bullet_{f_0}$ is the dualizing complex of $f$, i.e., it is (globally) quasi-isomorphic to the two-term complex, $$ NL^{\bullet}_{f_0}: \ f_0^*\Omega_{X_0/k} \to \Omega_{C_0/k}, $$ concentrated in degrees $-1$ and $0$.

In some form, this is described in Behrend-Fantechi, particularly the last few sections. I also recommend the first chapter of Kollár's textbook, "Rational Curves on Algebraic Varieties". Sernesi's book on Hilbert schemes is also great. Debarre's book is wonderful, and very readable, ...