The recent discovery of a Higgs boson at LHC, while establishing the Higgs mechanism as the way of electroweak symmetry breaking, started an era of precision measurements involving the Higgs boson. In an effective Lagrangian framework, we consider the [Formula: see text] process, at an ILC running at a center-of-mass energy of 500 GeV to investigate the effect of the [Formula: see text] and [Formula: see text] couplings on the sensitivity of [Formula: see text] coupling on this process. Our results show that the sensitivity of the trilinear Higgs self-couplings on this process has somewhat strong dependence on the Higgs-gauge boson couplings. Single- and two-parameter reaches of ILC with integrated luminosity of 1000 fb[Formula: see text] are obtained on all the effective couplings indicating how these limits are affected by the presence of anomalous [Formula: see text] and [Formula: see text] couplings. The kinematic distributions studied to understand the effect of the anomalous couplings, again, show strong influence of [Formula: see text]–[Formula: see text] couplings on the dependence of these distributions on [Formula: see text] coupling. Similar results are indicated in the case of the process, [Formula: see text], considered at a center-of-mass energy of 2 TeV, where the cross-section is large enough. The effect of [Formula: see text] and [Formula: see text] couplings on the sensitivity of [Formula: see text] coupling is clearly established through our analyses in this process.