Serious game-based learning is a high-fashioned topic in educational technology which helps children learn and train their skills. Most existing literature focuses on the improvement of children's specific abilities or skills, such as reading comprehension, attention, memory, etc. However, for cognitively impaired children, it is necessary to train their comprehensive cognitive ability. In this paper, serious game-based learning is defined as the integration of cognitive theory, instructional content and gaming elements to create educational games that leverage the motivational and immersive nature of gaming to enhance learning outcomes. We classify the gamification elements involved in cognitive training and then combine them with Piaget's theory of cognitive development in order to develop a cognitive training framework consists of four sessions which is expected to enhance the cognitive function of children with cognitive deficits. In total 50 cognitively impaired children are recruited to participate in the test. Evaluation results reveal that our cognitive training framework is helpful in improving the cognitive ability of children with cognitive deficits. However, the issue of how long this cognitive improvement will last still remains to be explored in future research.