Predicting Earth systems is an important yet challenging problem due to the high dimensionality, chaotic behaviour, and coupled dynamics of the ocean, atmosphere, and other subsystems of the Earth. Numerical models derived to predict these systems invariably contain model error due to incomplete domain knowledge, limited capabilities of representation, and unresolved processes due to spatial resolution. Hybrid modeling, the pairing of a physics-driven model with a data-driven component, has shown promise in outperforming both purely physics-driven and data-driven approaches in predicting complex systems. Here we demonstrate two new hybrid methods that combine temporal or spatiotemporal models with a data-driven component that may be modally decomposed to give insight into model error, or used to compensate a model during prediction. These techniques are demonstrated on two Earth system variables: coastal sea surface elevation and sea surface temperature, which highlight that the inclusion of the data-driven components increases the skill of predicting their evolution. Our work demonstrates that this hybrid approach may prove valuable for: improving models during model development, creating novel methods for data assimilation, and enhancing predictive accuracy when available models have significant structural error.