We compute the corrections to the two-loop amplitudes for gg → h, gg → hg, qg → hq, and qq¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ q\\overline{q} $$\\end{document} → hg due to a modified cubic Higgs self-coupling. The exact dependence on the Higgs and top-quark masses across the entire 2 → 2 phase space is determined by numerically solving a system of differential equations for the relevant master integrals. The calculated amplitudes are crucial for evaluating the impact of the considered corrections on exclusive pp → hj production at the hadronic event level. As an application, we calculate the non-universal, kinematic-dependent cubic Higgs self-coupling corrections to the Higgs-boson transverse momentum distribution in gluon-gluon fusion Higgs production for arbitrary values of the transverse momentum.