  The discovery of dark matter (DM) at XENONnT or LZ would place constraints on
DM particle mass and coupling constants. It is interesting to ask when these
constraints can be compatible with the DM thermal production mechanism. We
address this question within the most general set of renormalisable models that
preserve Lorentz and gauge symmetry, and that extend the Standard Model by one
DM candidate of mass $m_{\rm DM}$ and one particle of mass $M_{med}$ mediating
DM-quark interactions. Our analysis divides into two parts. First, we postulate
that XENONnT/LZ has detected $\mu_S\sim\mathcal{O}(100)$ signal events, and use
this input to calculate the DM relic density, $\Omega_{DM} h^2$. Then, we
identify the regions in the $M_{med} - \Omega_{DM} h^2$ plane which are
compatible with the observed signal and with current CMB data. We find that for
most of the models considered here, $\mathcal{O}(100)$ signal events at
XENONnT/LZ and the DM thermal production are only compatible for resonant DM
annihilations, i.e. for $M_{med}\simeq2 m_{DM}$. In this case, XENONnT/LZ would
be able to simultaneously measure $m_{DM}$ and $M_{med}$. We also discuss the
dependence of our results on $m_{DM}$, $\mu_S$ and the DM spin, and provide
analytic expressions for annihilation cross-sections and mediator decay widths
for all models considered in this study.
