Artificial Intelligence (AI) is increasingly being used in manufacturing to automate tasks and process data, leading to what has been termed Industry. 4.0. However, as we move towards Industry 5.0, there is a need to incorporate societal and human-centric dimensions into the development and deployment of AI software artefacts. This requires blending ethical considerations with existing practices and standards. To address this need, the TAI-PRM framework has been developed. It builds upon established methods, such as Failure Mode and Effect Analysis (FMEA) and the Industrial ISO 31000, to manage risks associated with AI artefacts in the manufacturing sector. The framework identifies ethical considerations as hazards that can impact system processes and sustainability and provides tools and metrics to manage these risks. To validate the framework, it was applied in an EU project for Digital Twins on AI for manufacturing. The results showed that TAI-PRM can effectively identify and track different failure modes associated with AI artefacts and help users to manage ethical risks associated with their deployment. By incorporating ethical considerations into risk management processes, the framework enables the developing and deploying trustworthy AI in the manufacturing sector. The online version contains supplementary material available at 10.1007/s43681-023-00417-y.