  The gradient discretisation method is a generic framework that is applicable
to a number of schemes for diffusion equations, and provides in particular
generic error estimates in $L^2$ and $H^1$-like norms. In this paper, we
establish an improved $L^2$ error estimate for gradient schemes. This estimate
is applied to a family of gradient schemes, namely, the Hybrid Mimetic Mixed
(HMM) schemes, and yields an $\mathcal O(h^2)$ super-convergence rate in $L^2$
norm, provided local compensations occur between the cell points used to define
the scheme and the centers of mass of the cells. To establish this result, a
modified HMM method is designed by just changing the quadrature of the source
term; this modified HMM enjoys a super-convergence result even on meshes
without local compensations. Finally, the link between HMM and Two-Point Flux
Approximation (TPFA) finite volume schemes is exploited to partially answer a
long-standing conjecture on the super-convergence of TPFA schemes.
