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We present an effective model for the one-dimensional Lyman-$α$ flux power spectrum far above the baryonic Jeans scale. The main new ingredient is constituted by a set of two parameters that encode the impact of small, highly non-linear scales on the one-dimensional power spectrum on large scales, where it is measured by BOSS. We show that, by marginalizing over the model parameters that capture the impact of the intergalactic medium, the flux power spectrum from both simulations and observations can be described with high precision. The model displays a degeneracy between the neutrino masses and the (unknown, in our formalism) normalization of the flux power spectrum. This degeneracy can be lifted by calibrating one of the model parameters with simulation data, and using input from Planck CMB data. We demonstrate that this approach can be used to extract bounds on the sum of neutrino masses with comparably low numerical effort, while allowing for a conservative treatment of uncertainties from the dynamics of the intergalactic medium. An explorative analysis yields an upper bound of $0.16\,$eV at $95\%$ C.L. when applied to BOSS data at $3\leq z\leq 4.2$. We also forecast that if the systematic and statistical errors will be reduced by a factor two the upper bound will become $0.1\,$eV at $95\%$ C.L., and $0.056\,$eV when assuming a $1\%$ error.