High-index Mie-resonant dielectric nanostructures provide a new framework to manipulate light at the nanoscale. In particular their local field confinement together with their inherently low losses at frequencies below their bandgap energy allows to efficiently boost and control linear and nonlinear optical processes. Here, nanoantennas composed of a thin indium-tin oxide (ITO) layer in the center of a dielectric gallium phosphide (GaP) nanodisc are investigated. While the linear response is similar to that of a pure GaP nanodisc, it is shown that second harmonic generation is enhanced across a broadband wavelength range. On the other hand, third harmonic generation is only marginally enhanced around the epsilon-near-zero wavelength of ITO. Linear and nonlinear finite-difference time-domain simulations show that despite the high refractive index contrast leading to strong field confinement inside the antenna's ITO layer, the nanogap enhancement effect is mitigated by the low nonlinear volume of the nanogap layer and the antenna's behavior at the harmonic wavelength. Measurement of ITO and GaP nonlinear susceptibilities additionally show a comparative advantage for harmonic generation in GaP. These investigations deliver insights on the mechanisms at play in nonlinear nanogap antennas and their potential applications as nanoscale devices.