  We study the regularity of minimizers for a variant of the soap bubble
cluster problem: \begin{align*}
  \min \sum_{\ell=0}^N c_{\ell} P( S_\ell)\,, \end{align*} where $c_\ell>0$,
among partitions $\{S_0,\dots,S_N,G\}$ of $\mathbb{R}^2$ satisfying $|G|\leq
\delta$ and an area constraint on each $S_\ell$ for $1\leq \ell \leq N$. If
$\delta>0$, we prove that for any minimizer, each $\partial S_{\ell}$ is
$C^{1,1}$ and consists of finitely many curves of constant curvature. Any such
curve contained in $\partial S_{\ell} \cap \partial S_{m}$ or $\partial S_\ell
\cap \partial G$ can only terminate at a point in $\partial G \cap \partial
S_\ell \cap \partial S_{m}$ at which $G$ has a cusp. We also analyze a similar
problem on the unit ball $B$ with a trace constraint instead of an area
constraint and obtain analogous regularity up to $\partial B$. Finally, in the
case of equal coefficients $c_\ell$, we completely characterize minimizers on
the ball for small $\delta$: they are perturbations of minimizers for
$\delta=0$ in which the triple junction singularities, including those possibly
on $\partial B$, are ``wetted" by $G$.
