  We study cover times of subsets of ${\mathbb Z}^2$ by a two-dimensional
massive random walk loop soup. We consider a sequence of subsets $A_n \subset
{\mathbb Z}^2$ such that $|A_n| \to \infty$ and determine the distributional
limit of their cover times ${\mathcal T}(A_n).$ We allow the killing rate
$\kappa_n$ (or equivalently the ``mass'') of the loop soup to depend on the
size of the set $A_n$ to be covered. In particular, we determine the limiting
behavior of the cover times for inverse killing rates all the way up to
$\kappa_n^{-1}=|A_n|^{1-8/(\log \log |A_n|)},$ showing that it can be described
by a Gumbel distribution. Since a typical loop in this model will have length
at most of order $\kappa_n^{-1/2}=|A_n|^{1/2},$ if $\kappa_n^{-1}$ exceeded
$|A_n|,$ the cover times of all points in a tightly packed set $A_n$ (i.e. a
square or close to a ball) would presumably be heavily correlated, complicating
the analysis. Our result comes close to this extreme case.
