This work addresses the problem of robot manipulation tasks under unknown dynamics, such as pick-and-place tasks under payload uncertainty, where active exploration and(/for) online parameter adaptation during task execution are essential to enable accurate model-based control. The problem is framed as dual control seeking a closed-loop optimal control problem that accounts for parameter uncertainty. We simplify the dual control problem by pre-defining the structure of the feedback policy to include an explicit adaptation mechanism. Then we propose two methods for reference trajectory generation. The first directly embeds parameter uncertainty in robust optimal control methods that minimize the expected task cost. The second method considers minimizing the so-called optimality loss, which measures the sensitivity of parameter-relevant information with respect to task performance. We observe that both approaches reason over the Fisher information as a natural side effect of their formulations, simultaneously pursuing optimal task execution. We demonstrate the effectiveness of our approaches for a pick-and-place manipulation task. We show that designing the reference trajectories whilst taking into account the control enables faster and more accurate task performance and system identification while ensuring stable and efficient control.