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We have witnessed the continuing arms race between backdoor attacks and the corresponding defense strategies on Deep Neural Networks (DNNs). Most state-of-the-art defenses rely on the statistical sanitization of the "inputs" or "latent DNN representations" to capture trojan behaviour. In this paper, we first challenge the robustness of such recently reported defenses by introducing a novel variant of targeted backdoor attack, called "low-confidence backdoor attack". We also propose a novel defense technique, called "HaS-Nets". "Low-confidence backdoor attack" exploits the confidence labels assigned to poisoned training samples by giving low values to hide their presence from the defender, both during training and inference. We evaluate the attack against four state-of-the-art defense methods, viz., STRIP, Gradient-Shaping, Februus and ULP-defense, and achieve Attack Success Rate (ASR) of 99%, 63.73%, 91.2% and 80%, respectively. We next present "HaS-Nets" to resist backdoor insertion in the network during training, using a reasonably small healing dataset, approximately 2% to 15% of full training data, to heal the network at each iteration. We evaluate it for different datasets - Fashion-MNIST, CIFAR-10, Consumer Complaint and Urban Sound - and network architectures - MLPs, 2D-CNNs, 1D-CNNs. Our experiments show that "HaS-Nets" can decrease ASRs from over 90% to less than 15%, independent of the dataset, attack configuration and network architecture.