  Using Poincar\'e series of $ K $-finite matrix coefficients of integrable
antiholomorphic discrete series representations of $ \mathrm{Sp}_{2n}(\mathbb
R) $, we construct a spanning set for the space $ S_\rho(\Gamma) $ of Siegel
cusp forms of weight $ \rho $ for $ \Gamma $, where $ \rho $ is an irreducible
polynomial representation of $ \mathrm{GL}_n(\mathbb C) $ of highest weight $
\omega\in\mathbb Z^n $ with $ \omega_1\geq\ldots\geq\omega_n>2n $, and $ \Gamma
$ is a discrete subgroup of $ \mathrm{Sp}_{2n}(\mathbb R) $ commensurable with
$ \mathrm{Sp}_{2n}(\mathbb Z) $. Moreover, using a variant of Mui\'c's integral
non-vanishing criterion for Poincar\'e series on unimodular locally compact
Hausdorff groups, we prove a result on the non-vanishing of constructed Siegel
Poincar\'e series.
