Cloud clutter is one of the main interfering factors for the detection of infrared small targets. Here, we propose a novel robust target detection method to eliminate cloud clutter interference and improve detection efficiency. Firstly, a divergence operator-based filter is proposed to enhance the local contrast around the target, highlighting the target area. Secondly, we introduce a constant false alarm rate criterion to detect the possible target areas. Finally, a nonlinear classifier based on kernel support vector machine (KSVM) is utilized to distinguish the target area from the possible areas. The optimal detection parameters are presented after the effect of different kernel functions of KSVM is shown. Eventually, comparative experiments show that the proposed method outperforms conventional methods in terms of detection probability, false alarm probability, and running time.