Let $C$ be an irreducible smooth project curve over $\mathbb C$ and $Y$ a variety over $\mathbb C$ locally of complete intersection. Write $Y^{\text sm}$ for the smooth locus of $Y$. Consider the 'space' of morphisms $f: C \rightarrow Y$ that 'generically take values in $Y^{\text sm}$', i.e. $f(c) \in Y^{\text sm}$ for all but finitely many points $c$ in $C$.

Does anyone know if this 'space' of morphisms is smooth or not? If the answer is yes, could you please suggest some references? If the answer is no, then is it possible to, say, impose extra conditions to make the 'space' smooth? Thank you very much for your help!