<p>We prove a quantitative inhomogeneous Hopf-Oleinik lemma for viscosity solutions of <Equation ID="Equ97"> <EquationSource Format="TEX">\( |\nabla u|^{\alpha }F(D^{2}u)=f \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msup> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> </msup> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>f</mi> </mrow> </math></EquationSource> </Equation>and, more generally, for viscosity supersolutions of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(|\nabla u|^{\alpha }\,\mathcal {M}^{-}_{\lambda ,\Lambda }(D^{2}u)\le f\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> </msup> <mspace width="0.166667em" /> <msubsup> <mrow> <mi mathvariant="script">M</mi> </mrow> <mrow> <mi>λ</mi> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> <mo>-</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation>. The result yields linear boundary growth with universal constants depending only on the structural data. As applications, we obtain Lipschitz regularity for viscosity solutions of one–phase Bernoulli free boundary problems driven by these degenerate fully nonlinear operators and derive <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varepsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>–uniform Lipschitz bounds for a one–phase flame propagation model.</p>