A major challenge in autonomous flights is unknown disturbances, which can jeopardize safety and cause collisions, especially in obstacle-rich environments. This letter presents a disturbance-aware motion planning and control framework for autonomous aerial flights. The framework is composed of two key components: a disturbance-aware motion planner and a tracking controller. The motion planner consists of a predictive control scheme and an online-adapted learned disturbance model. The tracking controller, developed using contraction control methods, ensures safety bounds on the quadrotor's behavior near obstacles with respect to the motion plan. The algorithm is tested in simulations with a quadrotor facing strong crosswind and ground-induced disturbances.