Nearly all classical inf-sup stable mixed finite element methods for the incompressible Stokes equations are not pressure-robust, i.e., the velocity error is dependent on the pressure. However, recent results show that pressure-robustness can be recovered by a non-standard discretization of the right hand side alone. This variational crime introduces a consistency error in the method which can be estimated in a straightforward manner provided that the exact velocity solution is sufficiently smooth. The purpose of this paper is to analyze the pressurerobust scheme with low regularity. The numerical analysis applies divergence-free H1-conforming Stokes finite element methods as a theoretical tool. As an example, pressure-robust velocity and pressure a-priori error estimates will be presented for the (first order) nonconforming CrouzeixRaviart element. A key feature in the analysis is the dependence of the errors on the Helmholtz projector of the right hand side data, and not on the entire data term. Numerical examples illustrate the theoretical results.