The quaternion MVDR (Q-MVDR) algorithm has better performance than the traditional complex-domain MVDR algorithm due to the inherent orthogonality between the dipole element components in quaternion signal model of the polarization sensitive array. However, when it comes to the situation where strong desired signal and steering vector are mismatched, the performance of Q-MVDR will be degraded significantly, which can even aggravate the signal self-nulling effect. To address this problem, a robust beamforming algorithm based on quaternion matrix reconstruction was proposed. First, a quaternion signal model of polarization sensitive array was established, and the covariance matrix reconstruction method was extended to the quaternion domain. The steering vector and power of interference signals were obtained by subspace method, and the Capon spectral estimator was adopted to reconstruct the interference-plus-noise covariance matrix. Then, according to the orthogonality of signal subspace and noise subspace, as well as the characteristics that the desired signal vector and signal subspace belong to the same subspace, an improved version of steering vector mismatch correction method was introduced, which utilized the projection technique of weight vector to eigenspace signal subspace. Finally, the effectiveness of the algorithm was verified by numerical simulations. Simulation results show that the proposed technique had better robustness than traditional Q-MVDR algorithm in effectively avoiding the performance degradation caused by signal self-nulling effect, and it is able to provide similar SINR close to the optimal value in both strong desired signal and steering vector mismatch cases.