Talbot-Lau x-ray imaging (TLXI) is an innovative and promising imaging technique providing information about the x-ray attenuation, scattering, and refraction features of objects. However, the method is susceptible to vibrations and system component imprecisions, which are inevitable in clinical and industrial practice. Those influences provoke grating displacements and hence errors in the acquired raw data, which cause moiré artifacts in the reconstructed images. We developed an enhanced reconstruction algorithm capable of compensating these errors by adjusting the grating positions and thus suppressing the occurrence of moiré artifacts. The algorithm has been developed with regard to a future application in medical practice. The capability of the algorithm is demonstrated on a medical data set of a human hand (post-mortem) acquired under clinical conditions using a pre-clinical TXLI prototype. It is shown that the algorithm reliably suppresses moiré artifacts, preserves image contrast, does not blur anatomical structures or prevent quantitative imaging, and is executable on low-dose data sets. In addition, the algorithm runs autonomously without the need of interaction or rework of the final results. In conclusion, the proposed reconstruction algorithm facilitates the use of TLXI in clinical practice and allows the exploitation of the method's full diagnostic potential in future medical applications.