  We present an effective model for the one-dimensional Lyman-$\alpha$ 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.
