Let $X$ be a compact complex surface and $u_1, u_2: \mathbb{P}^1 \longrightarrow X$ be rational curves that are not multiply covered that represents a class $\beta \in H_2(X, \mathbb{Z})$. Suppose $u_1$ is an immersion but $u_2$ is not an immersion, i.e. there exists some point $y \in \mathbb{P}^1$ such that $du_2|_y =0$. Doe this imply that there exists some $u_3: \mathbb{P}^1 \longrightarrow X$ such that $du_3|_y =0$ but any of the higher derivatives are non zero (i.e. $u_3$ has a genuine cusp at the point $y$ and not something more degenerate). Furthermore, $u_3$ should be an immersion at every other point. Note that I am assuming here that the $u_1$ exists, i.e. there is a curve that is an immersion (everywhere).
It seems to me that in a neighbourhood of $u_2$ such a $u_3$ should exist, but I don't see how to prove this.
