We use the Sherwood-Relics suite of hybrid hydrodynamical and radiative transfer simulations to model the effect of inhomogeneous reionisation on the 1D power spectrum of the Lyman-$alpha$ forest transmitted flux at redshifts $4.2leq z leq 5$. Relative to models that assume a homogeneous UV background, reionisation suppresses the power spectrum at small scales, $k sim 0.1rm,km^{-1},s$, by $sim 10$ per cent because of spatial variations in the thermal broadening kernel and the divergent peculiar velocity field associated with over-pressurised intergalactic gas. On larger scales, $k<0.03rm,km^{-1},s$, 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 reionisation must therefore be accounted for in analyses of forthcoming high precision measurements. We provide a correction for the Lyman-$alpha$ forest power spectrum at $4.1leq z leq 5.4$ in a form that can be easily applied within other parameter inference frameworks. 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 Lyman-$alpha$ forest data at $z>4$, $0.006 rm ,km^{-1},sleq k leq 0.2 rm, km^{-1},s$, we find inhomogeneous reionisation does not introduce any significant bias in thermal parameter recovery for the current measurement uncertainties of $sim 10$ per cent. However, for $5$ per cent uncertainties, $sim 1sigma$ shifts between the estimated and true parameters occur.
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