Machine Learning (ML) systems are becoming instrumental in the public sector, with applications spanning areas like criminal justice, social welfare, financial fraud detection, and public health. While these systems offer great potential benefits to institutional decision-making processes, such as improved efficiency and reliability, they still face the challenge of aligning intricate and nuanced policy objectives with the precise formalization requirements necessitated by ML models. In this paper, we aim to bridge the gap between ML and public sector decision-making by presenting a comprehensive overview of key technical challenges where disjunctions between policy goals and ML models commonly arise. We concentrate on pivotal points of the ML pipeline that connect the model to its operational environment, delving into the significance of representative training data and highlighting the importance of a model setup that facilitates effective decision-making. Additionally, we link these challenges with emerging methodological advancements, encompassing causal ML, domain adaptation, uncertainty quantification, and multi-objective optimization, illustrating the path forward for harmonizing ML and public sector objectives.