Most of the existing sparse recovery methods are based on the squared error criterion, i.e., ℓ₂-norm metric, by appropriately adding to a sparsity-promoting regularizer. This criterion is, however, statistically optimal only when the noise are Gaussian distributed. In fact, non-Gaussian impulsive noise with heavy tailed distribution has been reported in a variety of practical applications. To guarantee outlier-resistant sparse reconstruction for impulsive noise, in this paper we instead employ the generalized ℓ_p-norm (1 ≤ p <; 2) to quantify the residual error metric. By heuristically leveraging the sparsity-encouraging log-sum penalty, two iteratively reweighted algorithms are proposed for approximately solving the ℓ_p - ℓ₀ sparse recovery problem, where the reweighted matrices constructed from the previous iterative solution are considered both for ℓ_p and ℓ₀ metrics. Simulation results demonstrate the efficiency and robustness of the proposed algorithms.