In this paper, we examine the potential of Higgs boson production associated with a photon at the LHC to probe the new physics effects in the framework of the standard model effective field theory. It is shown that the differential kinematic distributions such as photon transverse momentum and invariant mass of Higgs+$γ$ in Higgs associated production are powerful variables to explore the coefficients of dimension six operators. The analysis is performed in the decay channel of Higgs boson into a $b\bar{b}$ pair including the main sources of background processes and a realistic simulation of the detector effects. We provide constraints at $95\%$ confidence level on the Wilson coefficients of dimension-six operators affecting Higgs boson plus a photon production. We show to what extent these limits could be improved at the high luminosity LHC. The effect of these constraints on a well-motivated beyond standard model scenario is presented. ; Comment: 7 figures, 3 tables