We establish the existence and uniqueness of solutions to an abstract nonlinear equation driven by a multiplicative noise of Levy type, which covers many hydrodynamical models including 2D Navier-Stokes equations, 2D MHD equations, the 2D Magnetic Bernard problem, and several Shell models of turbulence. In the existing litetrature on this topic, besides the classical Lipschitz and one sided linear growth conditions, other assumptions, which might be untypical, are also required on the coefficients of the stochastic perturbations. We do not require these untypical assumptions.