A molecular theoretical description is developed to describe the adsorption of nanoparticles (NPs) that are coated with polymers and functionalized with (surface) acid groups. Results are presented for the adsorption onto both negatively and positively charged surfaces as a function of pH and salt concentration, polymer coating, and NP size. An important finding is that nanoparticles that are coated with weak charge regulating acid molecules such as citric acid develop an asymmetric charge distribution close to a charged surface, due to their finite size. Depending on the sign of the surface charge of the adsorbing surface, a nanoparticle close to the surface either gains more charge or loses charge compared to its "bulk" degree of charge. This in turn influences the amount of NPs that adsorb. The effect of adsorption of negatively charged NPs onto a positively charged surface shows a nonmonotonical variation with pH. The described charging mechanism reveals that details such as size of the NP and acid distribution on the NP need to be considered to provide an accurate understanding of the adsorption process.