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Mobile manipulation in robotics is challenging due to the need of solving many diverse tasks, such as opening a door or picking-and-placing an object. Typically, a basic first-principles system description of the robot is available, thus motivating the use of model-based controllers. However, the robot dynamics and its interaction with an object are affected by uncertainty, limiting the controller's performance. To tackle this problem, we propose a Bayesian multi-task learning model that uses trigonometric basis functions to identify the error in the dynamics. In this way, data from different but related tasks can be leveraged to provide a descriptive error model that can be efficiently updated online for new, unseen tasks. We combine this learning scheme with a model predictive controller, and extensively test the effectiveness of the proposed approach, including comparisons with available baseline controllers. We present simulation tests with a ball-balancing robot, and door-opening hardware experiments with a quadrupedal manipulator.