Automated essay scoring systems typically rely on hand-crafted features to predict essay quality, but such systems are limited by the cost of feature engineering. Neural networks offer an alternative to feature engineering, but they typically require more annotated data. This paper explores network structures, contextualized embeddings and pre-training strategies aimed at capturing discourse characteristics of essays. Experiments on three essay scoring tasks show benefits from all three strategies in different combinations, with simpler architectures being more effective when less training data is available.