The rate for Higgs (H) bosons production in association with either one (tH) or two (tt^- H) top quarks is measured in final states containing multiple electrons, muons, or tau leptons decaying to hadrons and a neutrino, using proton–proton collisions recorded at a center-of-mass energy of 13Te by the CMS experiment. The analyzed data correspond to an integrated luminosity of 137fb^-1. The analysis is aimed at events that contain H → WW , H → ττ , or H → ZZ decays and each of the top quark(s) decays either to lepton+jets or all-jet channels. Sensitivity to signal is maximized by including ten signatures in the analysis, depending on the lepton multiplicity. The separation among tH , tt^-H , and the backgrounds is enhanced through machine-learning techniques and matrix-element methods. The measured production rates for the tt^-H and tH signals correspond to 0.92±0.19 (stat)_-0.13^+0.17 (syst) and 5.7±2.7 (stat)±3.0 (syst) of their respective standard model (SM) expectations. The corresponding observed (expected) significance amounts to 4.7 (5.2) standard deviations for tt^-H , and to 1.4 (0.3) for tH production. Assuming that the Higgs boson coupling to the tau lepton is equal in strength to its expectation in the SM, the coupling y_t of the Higgs boson to the top quark divided by its SM expectation, κ_t = y_t/y_t^SM, is constrained to be within -0.9 < κ_t < -0.7 or 0.7 < κ_t < 1.1 , at 95% confidence level. This result is the most sensitive measurement of the tt^-H production rate to date.