The purpose of this article is to give another molecular decomposition for members of weighted Hardy spaces, different from that given by Lee and Lin [J. Funct. Anal. 188 (2002), no. 2, 442–460], and to review some overlooked details. As an application of this decomposition, we obtain the boundedness on $H^{p}_{w}(\mathbb{R}^{n})$ of every bounded linear operator on some $L^{p_0}(\mathbb{R}^n)$ with $1 < p_0 < +\infty$, for all weights $w \in \mathcal{A}_{\infty}$ and all $0 < p \leq 1$ if $1< \frac{r_w -1}{r_w} p_0$, or all $0 < p < \frac{r_w -1}{r_w} p_0$ if $\frac{r_w -1}{r_w} p_0 \leq 1$, where $r_w$ is the critical index of $w$ for the reverse Hölder condition. In particular, the well-known results about boundedness of singular integrals from $H^{p}_w(\mathbb{R}^{n})$ into $L^{p}_{w}(\mathbb{R}^{n})$ and on $H^{p}_{w}(\mathbb{R}^{n})$ for all $w \in \mathcal{A}_{\infty}$ and all $0 < p \leq 1$ are established. We also obtain the $H^{p}_{w^{p}}(\mathbb{R}^{n})$-$H^{q}_{w^{q}}(\mathbb{R}^{n})$ boundedness of the Riesz potential $I_{\alpha}$ for $0 < p \leq 1$, $\frac{1}{q}=\frac{1}{p} - \frac{\alpha}{n}$, and certain weights $w$.