Plasmonic nanoparticles show highly sensitive optical properties upon local dielectric environment changes. Hybridisation of plasmonic nanoparticles with active polymeric materials can allow stimuli-responsive and multiplex sensing over conventional monotonic sensing capacity. Such heterogeneous adlayers around the plasmonic core component, however, are likely to perturb the local refractive index in the nanometre regime and lead to uncertainty in its intrinsic sensitivity. Herein we prepare a series of polystyrene-grafted polyhedral gold nanoparticles, cubic and concave cubic cores, with different edge lengths and polymer thicknesses with precise synthesis control. Their localised surface plasmon resonance (LSPR) spectral changes are monitored to understand the effect of core morphological details in the interplay of nanoscale polymeric shells. Quantitative image analysis of changes in the core and shell shape contours and finite-difference time-domain simulations of the corresponding LSPR spectra and electric field distributions reveal that the magnitude of the LSPR spectral shift is closely dependent on the core morphology, polymer shell thickness and electric field intensity. We also demonstrate that the polystyrene-grafted gold concave cube displays higher sensitivity for nanoscale refractive index change in the polymer shell than the polystyrene-grafted gold cube at different temperatures. Our systematic investigation will help design polymer-composited plasmonic nanosensors for desirable applications.