In this study, we introduce a novel model, termed the Weighted Basis Pursuit Dequantization ($ω$-BPDQ$_p$), which incorporates prior support information by assigning weights on the $\ell_1$ norm in the $\ell_1$ minimization process and replaces the $\ell_2$ norm with the $\ell_p$ norm in the constraint. This adjustment addresses cases where noise deviates from a Gaussian distribution, such as quantized errors, which are common in practice. We demonstrate that Restricted Isometry Property (RIP$_{p,q}$) and Weighted Robust Null Space Property ($ω$-RNSP$_{p,q}$) ensure stable and robust reconstruction within $ω$-BPDQ$_p$, with the added observation that standard Gaussian random matrices satisfy these properties with high probability. Moreover, we establish a relationship between RIP$_{p,q}$ and $ω$-RNSP$_{p,q}$ that RIP$_{p,q}$ implies $ω$-RNSP$_{p,q}$. Additionally, numerical experiments confirm that the incorporation of weights and the non-Gaussian constraint results in improved reconstruction quality.