When a very-low-absorbing sample is scanned at an x-ray computed tomography setup with a microfocus x-ray tube and a high-resolution detector, the obtained projection images contain not only absorption contrast but also phase contrast. While images without a phase signal can be reconstructed very well, such mixed phase and absorption images give rise to severe artifacts in the reconstructed slices. A method is described that applies a correction to these mixed projections to remove the phase signal. These corrected images can then be processed using a standard filtered backprojection algorithm to obtain reconstructions with only few or no phase artifacts. This new method, which we call the Bronnikov-aided correction (BAC), can be used in a broad variety of applications and without much additional effort. It is tested on a biological and a pharmaceutical sample, and results are evaluated and discussed by comparing them with those of conventional reconstruction methods.