  As weak lensing surveys are becoming deeper and cover larger areas,
information will be available on small angular scales down to the arcmin level.
To extract this extra information, accurate modelling of baryonic effects is
necessary. In this work, we adopt a baryonic correction model, which includes
gas both bound inside and ejected from dark matter (DM) haloes, a central
galaxy, and changes in the DM profile induced by baryons. We use this model to
incorporate baryons into a large suite of DM-only $N$-body simulations,
covering a grid of 75 cosmologies in the $\Omega_\mathrm{m}-\sigma_8$ parameter
space. We investigate how baryons affect Gaussian and non-Gaussian weak lensing
statistics and the cosmological parameter inferences from these statistics. Our
results show that marginalizing over baryonic parameters degrades the
constraints in $\Omega_\mathrm{m}-\sigma_8$ space by a factor of $2-4$ compared
to those with baryonic parameters fixed. We investigate the contribution of
each baryonic component to this degradation, and find that the distance to
which gas is ejected (from AGN feedback) has the largest impact due to its
degeneracy with cosmological parameters. External constraints on this
parameter, either from other datasets or from a better theoretical
understanding of AGN feedback, can significantly mitigate the impact of baryons
in an HSC-like survey.
