In this paper, a binaural beamforming algorithm for hearing aid applications is introduced. The beamforming algorithm is designed to be robust to some error in the estimate of the target speaker direction. The algorithm has two main components: a robust target linearly constrained minimum variance (TLCMV) algorithm based on imposing two constraints around the estimated direction of the target signal, and a post-processor to help with the preservation of binaural cues. The robust TLCMV provides a good level of noise reduction and low level of target distortion under realistic conditions. The post-processor enhances the beamformer abilities to preserve the binaural cues for both diffuse-like background noise and directional interferers (competing speakers), while keeping a good level of noise reduction. The introduced algorithm does not require knowledge or estimation of the directional interferers' directions nor the second-order statistics of noise-only components. The introduced algorithm requires an estimate of the target speaker direction, but it is designed to be robust to some deviation from the estimated direction. Compared with recently proposed state-of-the-art methods, comprehensive evaluations are performed under complex realistic acoustic scenarios generated in both anechoic and mildly reverberant environments, considering a mismatch between estimated and true sources direction of arrival. Mismatch between the anechoic propagation models used for the design of the beamformers and the mildly reverberant propagation models used to generate the simulated directional signals is also considered. The results illustrate the robustness of the proposed algorithm to such mismatches.