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Deep convolutional neural network (DCNN) models have been widely explored for skin disease diagnosis and some of them have achieved the diagnostic outcomes comparable or even superior to those of dermatologists. However, broad implementation of DCNN in skin disease detection is hindered by small size and data imbalance of the publically accessible skin lesion datasets. This paper proposes a novel single-model based strategy for classification of skin lesions on small and imbalanced datasets. First, various DCNNs are trained on different small and imbalanced datasets to verify that the models with moderate complexity outperform the larger models. Second, regularization DropOut and DropBlock are added to reduce overfitting and a Modified RandAugment augmentation strategy is proposed to deal with the defects of sample underrepresentation in the small dataset. Finally, a novel Multi-Weighted New Loss (MWNL) function and an end-to-end cumulative learning strategy (CLS) are introduced to overcome the challenge of uneven sample size and classification difficulty and to reduce the impact of abnormal samples on training. By combining Modified RandAugment, MWNL and CLS, our single DCNN model method achieved the classification accuracy comparable or superior to those of multiple ensembling models on different dermoscopic image datasets. Our study shows that this method is able to achieve a high classification performance at a low cost of computational resources and inference time, potentially suitable to implement in mobile devices for automated screening of skin lesions and many other malignancies in low resource settings.