  We consider the problem of performing connected sums in the context of
positive $k^{th}$ intermediate Ricci curvature. We show that such connected
sums are possible if the manifolds involved possess `$k$-core metrics' for some
$k$. Here, a $k$-core metric is a generalization of the notion of core metric
introduced by Burdick for positive Ricci curvature. Further, we show that
connected sums of linear sphere bundles over bases admitting such metrics admit
positive $k^{th}$ intermediate Ricci curvature for $k$ in a particular range.
This follows from a plumbing result we establish, which generalizes other
recent plumbing results in the literature and is possibly of independent
interest. As an example of a manifold admitting a $k$-core metric, we prove
that $\mathbb{H} P^n$ admits a $(4n-3)$-core metric and that $\mathbb{O}P^2$
admits a $9$-core metric, and we show that in both cases these are optimal.
