We study the problem of recovering sparse and compressible signals using a weighted ${\ell _p}$ minimization with $0 < p \leq 1$ from noisy compressed sensing measurements when part of the support is known a priori. To better model different types of non-Gaussian (bounded) noise, the minimization program is subject to a data-fidelity constraint expressed as the ${\ell _q}(2 \leq q < \infty)$ norm of the residual error. We show theoretically that the reconstruction error of this optimization is bounded (stable) if the sensing matrix satisfies an extended restricted isometry property. Numerical results show that the proposed method, which extends the range of $p$ and $q$ comparing with previous works, outperforms other noise-aware basis pursuit programs. For $p < 1$, since the optimization is not convex, we use a variant of an iterative reweighted ${\ell _2}$ algorithm for computing a local minimum.