In this paper, we consider a poverty alleviation (PA) agricultural product supply chain with an e-commerce platform, a farmer in poor areas, and a farmer in other general areas, where the platform with corporate social responsibility and consumers both have supporting preferences for poor areas. We investigate the optimal strategies by further considering logistics cost difference between the poor areas and other areas. The results indicate that the platform and consumers' supporting preferences are both beneficial to improve the poor farmer's profit, while they may decrease the sales quantity and the price of general agricultural products. In addition, the high logistics cost of poor areas has negative impacts on the sales of PA agricultural products and the poor farmer's profit. By virtue of some numerical analysis, we further discover that the platform's profit improves with its supporting preference only when consumers' supporting preference is relatively large, or when the platform and consumers' supporting preferences are both relatively small. In addition, the consumer surplus always improves with supporting preference.