We consider nonlinear stochastic evolution equations such as the stochastic p-Laplace equation with a nonlinear first-order perturbation. More precisely, the leading operator in our equation is a nonlinear, second order pseudomonotone operator of Leray-Lions type. In our setting, we may also add Lipschitz continuous perturbations of zero order. The multiplicative noise term is given by a stochastic integral with respect to a Q-Wiener process. We show well-posedness of the associated initial value problem for random initial data on a bounded domain with a homogeneous Dirichlet boundary condition. Therefore we consider a singular perturbation of our problem by a highter order operator. Through the a-priori etimates for the approximate solutions of the singular perturbation, only weak convergence is obtained. This convergence is not compatible with the nonlinearities in the equation. Therefore we use the theorems of Prokhorov and Skorokhod to establish existence of martingale solutions. Then, pathwise uniqueness follows from a L1-contraction principle and we may apply the method of Gyöngy-Krylov to obtain stochastically strong solutions. These well-posedness results serve as a basis for the study of variational inequalities and Lewy-Stampaccia inequalities for our problem.