This paper addresses the trajectory planning problem for search and coverage missions with an Unmanned Aerial Vehicle (UAV). The objective is to devise optimal coverage trajectories based on a utility map describing prior region information, assumed to be effectively approximated by a Gaussian Mixture Model (GMM). We introduce a Model Predictive Control (MPC) algorithm employing a relaxed formulation that promotes the exploration of the map by preventing the UAV from revisiting previously covered areas. This is achieved by penalizing intersections between the UAV's visibility regions along its trajectory. The algorithm is assessed in MATLAB and validated in Gazebo, as well as in outdoor experimental tests. The results show that the proposed strategy can generate efficient and smooth trajectories for search and coverage missions.