This question was answered by Misha in a comment.

Nothing useful can be said about the handle decomposition of $N$, because, among other things, if an $n$-manifold $N$ has boundary $M$, then the connect-sum of $N$ with any closed $n$-manifold still has boundary $M$. For example, given an arbitrary closed $5$-manifold $N$, we may remove a small $4$-ball to obtain a manifold with boundary $S^4$.