The rate for Higgs ( (H) ) bosons production in association with either one ( (tH) ) or two ( (t\overline{t}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 (13\phantom{\rule{0.166667em}{0ex}}\text{TeV}) by the CMS experiment. The analyzed data correspond to an integrated luminosity of 137 (\phantom{\rule{0.166667em}{0ex}}{\text{fb}}^{-1}). The analysis is aimed at events that contain (H\to WW), (H\to \tau \tau ), or (H\to 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), (t\overline{t}H), and the backgrounds is enhanced through machine-learning techniques and matrix-element methods. The measured production rates for the (t\overline{t}H) and (tH) signals correspond to (0.92±0.19\phantom{\rule{0.166667em}{0ex}}{\text{(stat)}}{-0.13}^{+0.17}\phantom{\rule{0.166667em}{0ex}}\text{(syst)}) and (5.7±2.7\phantom{\rule{0.166667em}{0ex}}\text{(stat)}±3.0\phantom{\rule{0.166667em}{0ex}}\text{(syst)}) of their respective standard model (SM) expectations. The corresponding observed (expected) significance amounts to 4.7 (5.2) standard deviations for (t\overline{t}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, ({\kappa }{t}={y}{t}/{y}{t}^{\mathrm{SM}}), is constrained to be within (-0.9<{\kappa }{t}<-0.7) or (0.7<{\kappa }_{t}<1.1), at 95% confidence level. This result is the most sensitive measurement of the (t\overline{t}H) production rate to date.