The non-Hermitian skin effect is a distinctive physical phenomenon that has attracted considerable attention in non-Hermitian systems. Investigating its inherent connection with the quantum control of non-Hermitian many-body systems is of significant theoretical importance. In this paper, we study a one-dimensional Fermi-Hubbard model with non-reciprocal hopping under open boundary conditions. By solving the corresponding Bethe ansatz equations, we obtain the exact solution of the system. On this basis, we further calculate the ground-state density distribution and momentum distribution, thereby achieving a systematic analysis of the non-Hermitian skin effect in a many-body system. Our results reveal a notable competition mechanism between interparticle interactions and the imaginary gauge phases. By adjusting the strength of interactions and the magnitude of the imaginary gauge field, the intensity of the non-Hermitian skin effect can be effectively modulated. The imaginary gauge field enhances this effect, whereas interactions exhibit a certain inhibitory influence. Together, they govern the evolution of the system's distribution in both real space and momentum space. Specifically, in the strongly repulsive regime at a particle density of one, the system exhibits a Mott insulating state, wherein each lattice site is singly occupied. Under these conditions, only a sufficiently strong imaginary gauge phase can induce a rightward accumulation of particles at the boundary. Conversely, for a filling configuration characterized by a total particle density of two and a spin down density of one, the sites are doubly occupied by particles of opposite spin. In this latter case, the occupation structure remains unaffected by the presence of the imaginary gauge phase. This study elucidates the regulatory mechanisms of interactions and non-Hermitian coupling on the skin effect, providing an important foundation for the study of non-Hermitian strongly correlated many-body systems.