This paper introduces a trajectory planning algorithm for search and coverage missions with an Unmanned Aerial Vehicle (UAV) based on an uncertainty map that represents prior knowledge of the target region, modeled by a Gaussian Mixture Model (GMM). The trajectory planning problem is formulated as an Optimal Control Problem (OCP), which aims to maximize the uncertainty reduction within a specified mission duration. However, this results in an intractable OCP whose objective functional cannot be expressed in closed form. To address this, we propose a Model Predictive Control (MPC) algorithm based on a relaxed formulation of the objective function to approximate the optimal solutions. This relaxation promotes efficient map exploration by penalizing overlaps in the UAV's visibility regions along the trajectory. The algorithm can produce efficient and smooth trajectories, and it can be efficiently implemented using standard Nonlinear Programming solvers, being suitable for real-time planning. Unlike traditional methods, which often rely on discretizing the mission space and using complex mixed-integer formulations, our approach is computationally efficient and easier to implement. The MPC algorithm is initially assessed in MATLAB, followed by Gazebo simulations and actual experimental tests conducted in an outdoor environment. The results demonstrate that the proposed strategy can generate efficient and smooth trajectories for search and coverage missions.