In this paper, a feedback training approach for efficiently dealing with distribution shift in synthetic aperture radar target detection using a Bayesian convolutional neural network is proposed. After training the network on in-distribution data, it is tested on out-of-distribution data. Samples that are classified incorrectly with high certainty are fed back for a second round of training. This results in the reduction of false positives in the out-of-distribution dataset. False positive target detections challenge human attention, sensor resource management, and mission engagement. In these types of applications, a reduction in false positives thus often takes precedence over target detection and classification performance. The classifier is used to discriminate the targets from the clutter and to classify the target type in a single step as opposed to the traditional approach of having a sequential chain of functions for target detection and localisation before the machine learning algorithm. Another aspect of automated synthetic aperture radar detection and recognition problems addressed here is the fact that human users of the output of traditional classification systems are presented with decisions made by “black box” algorithms. Consequently, the decisions are not explainable, even to an expert in the sensor domain. This paper makes use of the concept of explainable artificial intelligence via uncertainty heat maps that are overlaid onto synthetic aperture radar imagery to furnish the user with additional information about classification decisions. These uncertainty heat maps facilitate trust in the machine learning algorithm and are derived from the uncertainty estimates of the classifications from the Bayesian convolutional neural network. These uncertainty overlays further enhance the users’ ability to interpret the reasons why certain decisions were made by the algorithm. Further, it is demonstrated that feeding back the high-certainty, incorrectly classified out-of-distribution data results in an average improvement in detection performance and a reduction in uncertainty for all synthetic aperture radar images processed. Compared to the baseline method, an improvement in recall of 11.8%, and a reduction in the false positive rate of 7.08% were demonstrated using the Feedback-assisted Bayesian Convolutional Neural Network or FaBCNN.