To address the need for mission re-planning in the event of thrust failures or other emergencies during the powered flight phase of a launch vehicle, this paper designs an online trajectory reconstruction method based on pseudospectral-sequential convex optimization. By combining pseudospectral discretization with convex optimization, the method improves convergence accuracy and efficiency compared to traditional convex optimization. It also transforms non-convex orbital element constraints into convex constraints, reducing problem complexity. Additionally, an adaptive trust region adjustment strategy is designed to accelerate convergence. Using the proposed method, trajectory re-planning is performed for single-stage and dual-stage rocket thrust failure scenarios, with the optimization objective of minimizing fuel consumption. The solution efficiency is on the order of hundreds of milliseconds, enabling online trajectory planning for the powered flight phase of launch vehicles.