In this paper, we introduced a new class of weak Hardy spaces, denoted by $H_b^{p,\mathrm{\infty }}$, and provided an analysis of their atomic decomposition. As an application, we established the boundedness of Calderón-Zygmund operators (CZOs) from $H^p$ to $H_b^{p,\mathrm{\infty }}$ including cases at the critical exponent

$$p=\frac{n}{n+\delta },$$

where $\delta$ represents the regularity index of the distributional kernel. Moreover, the boundedness of CZOs from $H^{p,\mathrm{\infty }}$ to $H_b^{p,\mathrm{\infty }}$ was demonstrated for

$$\frac{n}{n+\delta }<p\le 1.$$