This paper presents a rapid ascent trajectory planning method for a rocket-based combined-cycle powered aerospace plane subject to terminal point deviation caused by mission uncertainties. The main feature lies in that the proposed algorithm achieves rapid control updates and accurate trajectory tracking by incorporating the Legendre pseudospectral convex optimization and the improved model predictive static programming. Consequently, terminal point deviation and process inequality constraints are effectively handled even during multiple propulsion mode transitions. The introduced Legendre pseudospectral convex optimization method for trajectory initialization generates constraint-feasible initial trajectories across operational modes, thereby improving convergence and solution quality, while the enhanced Model Predictive Static Programming algorithm rapidly updates control commands and maintains trajectory accuracy. Numerical simulations demonstrate the effectiveness of the proposed approach in ensuring precise terminal guidance under mission uncertainties.