Scanning transmission X-ray microscopy (STXM) produces images in which each pixel value is related to the measured attenuation of an X-ray beam. In practice, the location of the illuminated region does not exactly match the desired uniform pixel grid. This error can be measured using an interferometer. In this paper, we propose a spline-based reconstruction method for STXM which takes these position errors into account. We achieve this by formulating the reconstruction problem as a continuous-domain inverse problem in a spline basis, and by using Hessian nuclear-norm regularization. We solve this problem using the standard ADMM algorithm, and we demonstrate the pertinence of our approach on both simulated and real STXM data.