  We implement the momentum dissipation introduced by spatial linear axionic
fields in a holographic model without self-duality, broke by Weyl tensor
coupling to Maxwell field, and study its response. It is found that for the
positive Weyl coupling parameter $\gamma>0$, the momentum dissipation
characterized by parameter $\hat{\alpha}$ drives the boundary conformal field
theory (CFT), in which the conductivity exhibits a peak at low frequency, into
the incoherent metallic phase with a dip, which is away from CFT due to the
introduction of axionic fields. While for $\gamma<0$, an oppositive scenario is
found. Our present model provides a possible route toward the problem that
which sign of $\gamma$ is the correct description of the CFT of boson Hubbard
model. In addition, we also investigate the DC conductivity, diffusion constant
and susceptibility. We find that for each of these observables there is a
specific value of $\hat{\alpha}$, for which these observables are independent
of $\gamma$. Finally, the electromagnetic (EM) duality is also studied and we
find that there is also a specific value of $\hat{\alpha}$, for which the
particle-vortex duality related by the change of the sign of $\gamma$ in the
boundary theory holds better than for other values of $\hat{\alpha}$.
