  We construct a modular desingularisation of
$\overline{\mathcal{M}}_{2,n}(\mathbb{P}^r,d)^{\text{main}}$. The geometry of
Gorenstein singularities of genus two leads us to consider maps from prestable
admissible covers: with this enhanced logarithmic structure, it is possible to
desingularise the main component by means of a logarithmic modification. Both
isolated and non-reduced singularities appear naturally. Our construction gives
rise to a notion of reduced Gromov-Witten invariants in genus two.
