Smoothing in Naive Bayes model
The typical reason for smoothing in the first place is to handle cases where $\#\{X_i = x_i | Y = y\} = 0$. If this wasn't done, we would always get $P(Y=y|X=x) = 0$ whenever this was the case. 

This happens when, for example, classifying text documents you encounter a word that wasn't in your training data, or just didn't appear in some particular class.

On the other hand, in the case of the class prior probability, $P(Y = y)$, this situation should not occur. If it did this would mean you are trying to assign objects to classes which didn't even appear in the training data.

Also, I've never encountered the term $k$-smoothing. Laplace or Additive smoothing is much more common.