  Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $\varphi:\
\Omega\times[0,\infty)\to [0,\infty)$ be a Musielak--Orlicz function. In this
article, the authors establish the atomic characterizations of weak martingale
Musielak--Orlicz Hardy spaces $WH_{\varphi}^s(\Omega)$,
$WH_{\varphi}^M(\Omega)$, $WH_{\varphi}^S(\Omega)$, $WP_{\varphi}(\Omega)$ and
$WQ_{\varphi}(\Omega)$. Using these atomic characterizations, the authors then
obtain the boundedness of sublinear operators from weak martingale
Musielak--Orlicz Hardy spaces to weak Musielak--Orlicz spaces, and some
martingale inequalities which further clarify the relationships among
$WH_{\varphi}^s(\Omega)$, $WH_{\varphi}^M(\Omega)$, $WH_{\varphi}^S(\Omega)$,
$WP_{\varphi}(\Omega)$ and $WQ_{\varphi}(\Omega)$. All these results improve
and generalize the corresponding results on weak martingale Orlicz--Hardy
spaces. Moreover, the authors also improve all the known results on weak
martingale Musielak--Orlicz Hardy spaces. In particular, both the boundedness
of sublinear operators and the martingale inequalities, for the weak weighted
martingale Hardy spaces as well as for the weak weighted martingale
Orlicz--Hardy spaces, are new.
