A spectral problem, the x-derivative part of which is a simple generalization of the standard Ablowitz-Kaup-Newell-Segur and Kaup-Newell spectral problems, is presented with its associated generalized mixed nonlinear Schrödinger (GMNLS) model. The N-fold Darboux transformation with multi-parameters for the spectral problem is constructed with the help of gauge transformation. According to the Darboux transformation, the solution of the GMNLS model is reduced to solving a linear algebraic system and two first-order ordinary differential equations. As an example of application, we list the modulus formulae of the envelope one- and two-soliton solutions. Note that our model is a generalized one with the inclusion of four coefficients (a, b, c, and d), which involves abundant NLS-type models such as the standard cubic NLS equation, the Gerdjikov-Ivanov equation, the Chen-Lee-Liu equation, the Kaup-Newell equation, and the mixed NLS of Wadati and/or Kundu, among others.