In this study, we introduce a data-driven machine learning framework for improving the accuracy of wind plant flow models by learning turbulence model corrections based on data from higher-fidelity simulations. First, a high-dimensional PDE-constrained optimization problem is solved using gradient-based optimization with adjoints to determine optimal eddy viscosity fields that improve the agreement of a medium-fidelity Reynolds-Averaged Navier Stokes (RANS) model with large eddy simulations (LES). A supervised learning problem is then constructed to find general, predictive representations of the optimal turbulence closure. A machine learning technique using Gaussian process regression is trained to predict the eddy viscosity field based on local RANS flow field information like velocities, pressures, and their gradients. The Gaussian process is trained on LES simulations of a single turbine and implemented in a wind plant simulation with 36 turbines. We show improvement over the baseline RANS model with the machine learning correction, and demonstrate the ability to provide accurate confidence levels for the corrections that enable future uncertainty quantification studies.