Nickel cobalt-metal–organic framework (NiCo-MOF), with a semihollow spherical morphology composed of rhombic dodecahedron nanostructures, was synthesized using a scalable and facile wet chemical route. Such a structure endowed the material with open pores, which enabled rapid ion ingress and egress, and the high effective surface area of the MOF allowed the uptake and release of a large number of electrolyte ions during charge–discharge. By combining this NiCo-MOF cathode with a highly porous carbon (PC) anode (derived from the naturally grown and abundantly available bio-waste, namely, palm kernel shells), the resulting PC//NiCo-MOF supercapacitor using an aqueous potassium hydroxide (KOH) electrolyte delivered a capacitance of 134 F g–1, energy and power densities of 24 Wh kg–1 and 0.8 kW kg–1 at 1 A g–1, respectively, over an operational voltage window of 1.6 V. By employing thin interlayers of PC coated over a Whatman filter paper (PC@FP), the modified supercapacitor configuration of PC/PC@FP//PC@FP/NiCo-MOF delivered greatly enhanced performance. This cell delivered a capacitance of 520 F g–1 and an energy density of 92 Wh kg–1, improved by nearly 4-fold, compared to the analogous supercapacitor without the interlayers (at the same power and current densities and voltage window), thus evidencing the role of the cost-effective, electrically conducting porous carbon interlayers in amplifying the supercapacitor’s energy storage capabilities. Further, illumination of white light-emitting diodes (LEDs) using a three-series configuration and the photocharging of this supercapacitor with a solution-processed solar cell are also demonstrated. The latter confirms its ability to function as a stand-alone power supply system for electronic/computing devices, which can even operate under medium lighting conditions.