This paper contributes a method to design a novel navigation planner exploiting a learning-based collision prediction network. The neural network is tasked to predict the collision cost of each action sequence in a predefined motion primitives library in the robot's velocity-steering angle space, given only the current depth image and the estimated linear and angular velocities of the robot. Furthermore, we account for the uncertainty of the robot's partial state by utilizing the Unscented Transform and the uncertainty of the neural network model by using Monte Carlo dropout. The uncertainty-aware collision cost is then combined with the goal direction given by a global planner in order to determine the best action sequence to execute in a receding horizon manner. To demonstrate the method, we develop a resilient small flying robot integrating lightweight sensing and computing resources. A set of simulation and experimental studies, including a field deployment, in both cluttered and perceptually-challenging environments is conducted to evaluate the quality of the prediction network and the performance of the proposed planner.