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Balanced Graph Partitioning is a classical optimization problem where quality guarantees are computationally infeasible, and practical solvers therefore rely on manually engineered heuristics. Yet, the problem has also proven difficult for approaches that rely heavily on machine learning - especially since applications often need to partition graphs of huge scale in a short amount of time. Instead, we demonstrate how to achieve practical improvements with a more careful approach that uses machine learning to improve heuristic decisions within the state-of-the-art solver Mt-KaHyPar. We use a pre-trained neural network to predict a score for each edge, which then guides clustering decisions in the first phase of the partitioning (the coarsening). Combined with corresponding adjustments to the clustering algorithm and an efficient implementation of the neural network logic, we improve the overall solution quality while preserving the efficiency and scalability of the original algorithm. Our detailed evaluation on more than 180 graphs shows an average quality improvement of 2% on a class of graphs with beneficial properties, and unchanged quality on all remaining graphs. Moreover, our improvements generalize to a set of instances from the literature that are much larger than the graphs used during training.