The ongoing electrification of many energy systems has created a large demand for low-cost and scalable electrical energy storage solutions. Conducting polymer supercapacitors have received significant attention for this purpose due to the abundance of their constituent materials. Although there exists a large body of experimental work on conducting polymer supercapacitors, a detailed understanding of the mixed electronic-ionic transport processes within these devices and the included materials, is still lacking. Modelling, in combination with experimental data, is a powerful tool to facilitate a detailed understanding of the transport processes within the materials and devices. However, to date, there has been a shortage of physical models which account for the non-ideal capacitances typically found in conducting polymer-based supercapacitors. Here, we report a novel model which reproduces experimental data and provides insights into the cyclic voltammograms, galvanostatic charge-discharge curves, self-discharge characteristics, and impedance spectroscopy results of supercapacitors based on the conducting polymer poly(3,4-ethylenedioxythiophene):poly(styrene sulfonate) (PEDOT:PSS) and cellulose nanofibrils. We find that the non-ideal capacitive characteristics of the supercapacitors can be reproduced by the incorporation of heterogeneous ion transport features within the electrodes, comprising low ion diffusivity regions. The difference in charging rates of the high and low ion diffusivity regions accounts for the experimentally observed trends in cyclic voltammograms and self-discharge characteristics. The developed model demonstrates how complex transport processes, which govern the specifications of organic energy devices, can be analysed beyond the scope of conventional equivalent circuit models. It also provides an insight into how various transport and polarization processes are manifested in real measurement data and thus defines the limiting processes of conducting polymer energy storage devices.