Connected sum of one manifold WITH boundary with a manifold WITHOUT boundary

The result of your hybrid operation carried out on $\mathcal M$ and $\mathcal N$ is homeomorphic to the ordinary connected sum $\mathcal M \# \mathcal N$. Think of the balls removed from $M$ and $N$ as two cubes instead, say $B_M \subset M$ and $B_N \subset N$. Now let $B'_N \subset N$ be a larger cube which contains $B_N$ concentrically in its interior. You can then rewrite your connected sum like this: after removing $B_M$ and $B_N$, in addition remove the "cubical shell" $B'_n - B_N$ and glue that into $M$, resulting in what you get from $M$ by removing the interior of a ball contained entirely in the interior of $M$.