Extreme cornering in racing often leads to large sideslip angles, presenting a significant challenge for vehicle control. Conventional vehicle controllers struggle to manage this scenario, necessitating the use of a drifting controller. However, the large sideslip angle in drift conditions introduces model mismatch, which in turn affects control precision. To address this issue, we propose a model correction drift controller that integrates Model Predictive Control (MPC) with Gaussian Process Regression (GPR). GPR is employed to correct vehicle model mismatches during both drift equilibrium solving and the MPC optimization process. Additionally, the variance from GPR is utilized to actively explore different cornering drifting velocities, aiming to minimize trajectory tracking errors. The proposed algorithm is validated through simulations on the Simulink-Carsim platform and experiments with a 1:10 scale RC vehicle. In the simulation, the average lateral error with GPR is reduced by 52.8% compared to the non-GPR case. Incorporating exploration further decreases this error by 27.1%. The velocity tracking Root Mean Square Error (RMSE) also decreases by 10.6% with exploration. In the RC car experiment, the average lateral error with GPR is 36.7% lower, and exploration further leads to a 29.0% reduction. Moreover, the velocity tracking RMSE decreases by 7.2% with the inclusion of exploration.