We use the Sherwood-Relics suite of hybrid hydrodynamical and radiative transfer simulations to model the effect of inhomogeneous reionization on the 1D power spectrum of the Lyman alpha (Ly alpha) forest transmitted flux at redshifts 4.2 <= z <= 5. Relative to models that assume a homogeneous ultraviolet background, reionization suppresses the power spectrum at small scales, k similar to 0.1 km(-1) s, by similar to 10 per cent because of spatial variations in the thermal broadening kernel and the divergent peculiar velocity field associated with overpressurized intergalactic gas. On larger scales, k < 0.03 km(-1) s, the power spectrum is instead enhanced by 10-50 per cent by large-scale spatial variations in the neutral hydrogen fraction. The effect of inhomogeneous reionization must therefore be accounted for in analyses of forthcoming high precision measurements. We provide a correction for the Ly alpha forest power spectrum at 4.1 <= z <= 5.4 that can be easily applied within other parameter inference frameworks using similar reionization models. We perform a Bayesian analysis of mock data to assess the extent of systematic biases that may arise in measurements of the intergalactic medium if ignoring this correction. At the scales probed by current high-resolution Ly alpha forest data at z > 4, 0.006 <= k <= 0.2 km(-1) s, we find inhomogeneous reionization does not introduce any significant bias in thermal parameter recovery for the current measurement uncertainties of similar to 10 per cent. However, for 5 per cent uncertainties, similar to 1 sigma shifts between the estimated and true parameters occur.
