What I'm looking for is some sort of 'Bertini-theorem' for curves. Let $X$ be a Calabi-Yau manifold and let $C$ be a union of rational curves on $X$. Are there any techniques or results that would allow me to conclude that (perhaps some multiple of) $C$ can be deformed into an irreducible curve?

Edit: I'm looking for *conditions* on $C$ so that it moves. For example, if $C$ is a divisor on a K3, and $C^2>0$, then a general element in the linear system of some power is smooth by the usual Bertini.