Harmonic Weak Siegel Maa{ss} Forms I

Given a non-holomorphic Saito-Kurokawa lift we construct a preimage under the vector-valued lowering operator. In analogy with the case of harmonic weak elliptic Maa{\ss} forms, this preimage allows for a natural decomposition into a meromorphic and a non-holomorphic part. In this way every harmonic weak Siegel Maa{\ss} form gives rise to a Siegel mock modular form.