This work considers a collocated radar scenario where a probing signal is emitted toward the targets of interest and records the received echoes. Estimating the relative delay-Doppler shifts of the targets allows determining their relative locations and velocities. However, the received radar measurements are often affected by impulsive non-Gaussian noise which makes a few measurements partially corrupted. While demixing radar signal and impulsive noise is challenging in general by traditional subspace-based methods, atomic norm minimization (ANM) has been recently developed to perform this task in a much more efficient manner. Nonetheless, the ANM cannot identify close delay-Doppler pairs and also requires many measurements. Here, we propose a smoothed l(0) atomic optimization problem encouraging both the sparse features of the targets and the impulsive noise. We design a majorization-minimization algorithm that converges to the solution of the proposed non-convex problem using alternating direction method of multipliers (ADMM). Simulations results verify the superior accuracy of our proposed algorithm even for very close delay-Doppler pairs in comparison to ANM with around 40 dB improvement.