  The weak charge of the proton $Q_{\textrm{w}}$ is one of the most fundamental
quantities in physics. It can be determined by measuring the parity asymmetry
$A_{\textrm{PV}}$ in elastic $ep$ scattering, where the $\gamma Z$-exchange
contributions are crucial. For the past fifteen years, dispersion relations
(DRs) in the forward limit have been widely used as a model-independent method
to estimate these contributions. In this work, we study corrections to these
forward-limit DRs. We first estimate these corrections using pointlike
interactions as an illustrative example. We then estimate the $\gamma
Z$-exchange contributions for the upcoming P2 experiment through both direct
calculation and the forward-limit DRs, within the framework of low-energy
effective interactions. The results indicate that the correction to the
forward-limit DR for $\Box_{\gamma Z}^{V}$ is around 47\% for the upcoming P2
experiment, which will significantly modify the extracted value of
$Q_{\textrm{w}}$.
