Abstract: We denote by $$\overline{\mathcal{H}}_g$$ the closure of the hyperelliptic locus in the moduli space of stable curves of genus g. We consider the map $$\operatorname{Sym}^2(\operatorname{Pic}(\overline{\mathcal{H}}_g)) \to \mathrm{CH}^2(\overline{\mathcal{H}}_g)$$ and prove the kernel of the map is generated by a single relation. Moreover, the relation depends on the parity of g, but otherwise the relation has a simple recursive form.