Let $(X,0)$ be a normal surface singularity. Suppose that it does not admit a smoothing.

Is it possible that there exists an isolated surface singularity $(Y,0)$ reduced near $0$ which is not irreducible and $(X,0)$ is one of its irreducible components and such that $(Y,0)$ does admit a smoothing? Is there some obstruction for this to happen?

Are these kind of questions treated somewhere?