Numerical simulations are used to study the interaction dynamics between pump spectral broadening and third harmonic generation during supercontinuum generation in a highly nonlinear fused silica nanowire using sub-100 fs pump pulses. In particular, we show that the superposition of the generated third harmonic with the spectrally-broadened pump pulse exhibits significant dependence on the input pump carrier envelope phase, and we examine this dependence as a function of pulse duration and propagation distance.