This paper addresses an enriched nonconforming Multiscale Finite Element Method (MsFEM) to solve viscous incompressible flow problems in genuine heterogeneous or porous media. In the work of [B. P. Muljadi, J. Narski, A. Lozinski, and P. Degond, Multiscale Modeling \& Simulation 2015 13:4, 1146-1172] and [G. Jankowiak and A. Lozinski, arXiv:1802.04389 [math.NA], 2018], a nonconforming MsFEM has been first developed for Stokes problems in such media. Based on these works, we propose an innovative enriched nonconforming MsFEM where the approximation space of both velocity and pressure are enriched by weighting functions which are defined by polynomials of higher-degree. Numerical experiments show that this enriched nonconforming MsFEM improves significantly the accuracy of the nonconforming MsFEMs. Theoretically, this method provides a general framework which allows to find a good compromise between the accuracy of the method and the computing costs, by varying the degrees of polynomials.