Accurate heat flux prediction is a key problem in the numerical simulation of hypersonic flows. When the singularity axis appears in the grid， it further increases the difficulty of accurate flow simulation and heat flux calculation. Aiming at the problems of instability of flow structure and large errors of heat flux prediction in the finite difference calculation of singularity axis problems， a hybrid algorithm for solving the singularity axis problem with high-precision difference schemes is proposed by combining the flux reconstruction and conservation reconstruction. The algorithm is based on conservative variables reconstruction near the singularity axis， which can reflect the influence of the flow field distribution near the singularity axis. The algorithm can effectively solve the singularity axis problem in numerical simulation. Applying the hybrid algorithm with WENO3-JS and WENO3-PRM1,12 scheme， the numerical simulations of hypersonic flow around sphere， HB-2， sphere cone and Apollo are carried out. The results show that the new method can effectively solve the problems of flow instability with singularity axis， and can achieve accurate heat flux prediction.