For any pointed space $X$, the mod $p$ Lannes-Zarati homomorphism $\varphi_s^{\widetilde{H}^*X}$ is considered as a graded associated version of the mod $p$ Hurewicz map $h_*:\pi_*(QX)\to H_*QX$ in the $E_2$-term of the Adams spectral sequence. In this paper, we investigate the behavior of $\varphi_3^{\fp}$ and $\varphi_s^{\widetilde{H}^*(B\mathbb{Z}/p)}\ (s\leq1)$ for an odd prime $p$.