Context. Parker Solar Probe (PSP) observations have revealed that most of the solar wind acceleration occurs very close to the Sun. This acceleration is partly due to the global electric potential originating from the mass disparity between electrons and protons, coupled with the constraints of charge quasi-neutrality and zero-current conditions in the solar wind plasma. However, the exact mechanism that accounts for the remaining acceleration has not yet been identified. Aims. We aim to provide a framework that incorporates the electric-field-driven component of the acceleration while also introducing an additional acceleration mechanism via a velocity-space diffusion of the particles. This will help us determine the extent of extra acceleration, beyond the electric-field-driven component, required to fully reproduce the acceleration of the solar wind in theoretical models. Methods. We modified an existing kinetic exospheric model to account for the unexplained solar wind acceleration by including velocity-space diffusion, thereby capturing the effect of collisions and wave-particle interactions within the exospheric approach. We compared the electric field derived from the sunward deficit of velocity distribution functions observed by PSP between 13.3 and 50 solar radii (Rs) with the electric field found self-consistently by the kinetic exospheric model. Results. The effect of velocity-space diffusion is found to reduce the temperature anisotropy and impact the solar wind acceleration while leaving the electric potential unchanged. The approach described in this work enables the diffusion to be adjusted to effectively reduce or increase the solar wind acceleration. Even without diffusion, the model is able to reproduce the anticorrelation between the electric potential and the solar wind terminal velocity found by PSP. This suggests that the electric potential might still be of major importance in explaining the solar wind acceleration.