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We consider a class of nonlinear saddle point problems with various applications in PDEs and optimal control problems and propose an algorithmic framework based on some inexact Uzawa methods in the literature. Under mild conditions, the convergence of this algorithmic framework is uniformly proved and the linear convergence rate is estimated. We take an elliptic optimal control problem with control constraints as an example to illustrate how to choose application-tailored preconditioners to generate specific and efficient algorithms by the algorithmic framework. The resulting algorithm does not need to solve any optimization subproblems or systems of linear equations in its iteration; each of its iterations only requires the projection onto a simple admissible set, four algebraic multigrid V-cycles, and a few matrix-vector multiplications. Its numerical efficiency is then demonstrated by some preliminary numerical results. [ABSTRACT FROM AUTHOR]