HONG KONG—China’s Communist Party said it is investigating alleged wrongdoing by an influential businessman who has been an outspoken critic of President Xi Jinping, signaling harsh punishment for the author of a scathing essay decrying the Chinese leader’s handling of the coronavirus pandemic.

The probe against Ren Zhiqiang, a real-estate mogul and a well-connected Communist Party member, comes as Beijing trumpets its success in containing the coronavirus and seeks to quell public anger around initial government missteps.

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