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In this article, we provide a methodology to reconstruct high-Reynolds number turbulent mean-flows from few time-averaged measurements. A turbulent flow over a backward-facing step at Re = 28275 is considered to illustrate the potential of the approach. The data-assimilation procedure, based on a variational approach, consists in correcting a given baseline model by tuning space-dependent source terms such that the corresponding solution matches available measurements (obtained here from direct-numerical simulations). The baseline model chosen here consists in Reynolds-Averaged-Navier-Stokes (RANS) equations closed with the turbulence Spalart-Allmaras model. We investigate two possible tuning functions: a source term in the momentum equations, which is able to compensate for the deficiencies in the modeling of the Reynolds stresses by the Boussinesq approximation and a source term in the turbulence equation, which modifies the balance between the eddy-viscosity production and dissipation. The quality of the mean-flow reconstruction strongly depends on the baseline model and on the quantity of measurements. In the case of many measurements, very accurate reconstructions of the mean-flow are obtained with the model corrected by the source term in the momentum equations, while the reconstruction is more approximate when tuning the source term in the turbulence model. In the case of few measurements, this "rigidity" of the corrected turbulence model is favourably used and allows the best mean-flow reconstruction. The flexibility / rigidity of a model is further discussed in the light of a singular-value decomposition of the linear input / output operator between source term and measurements.