Steering vector (SV) mismatch caused by the DOA uncertainty in the source leads to a remarkable performance degradation for adaptive beamforming particularly in situation where the training data includes an unknown expected signal (ES) component. To mitigate this problem, a robust adaptive beamforming method based on SV phase correction and covariance matrix reconstruction is proposed in this letter. The first step is to correct the SV phase of the ES using the maximum a posteriori (MAP) estimation method. Next, the Gauss-Chebyshev quadrature is introduced to efficiently reconstruct the interference-plus-noise covariance matrix by integrating within the corrected azimuthal sector. The effectiveness and superiority of the proposed method in mitigating SV mismatch errors are confirmed by both numerical simulations and experimental results.