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The $^{12}{\rm C}(\alpha,\gamma)^{16}$O reaction, determining the survival ofcarbon in red giants, is of interest for nuclear reaction theory and nuclearastrophysics. A specific feature of the $^{16}$O nuclear structure is thepresence of two subthreshold bound states, (6.92 MeV, 2$^+$) and (7.12 MeV,1$^-$), that dominate the behavior of the low-energy $S$-factor. The strengthof these subthreshold states is determined by their asymptotic normalizationcoefficients (ANCs), which need to be known with high accuracy. Recently, usinga model-independent extrapolation method, Blokhintsev {\it et al.} [Eur. Phys.J. A {\bf 59} (2023) 162] determined the ANCs for the $\alpha$-particle removaltaking into account three subthreshold states in $^{16}$O. The goal of thispaper is to address four main problems elucidating the impact of thesubthreshold ANCs on the low-energy $S$-factor. Firstly, we analyse theconnection between variations of the subthreshold ANCs and the low-energy$S$-factor, in particular, at the most effective energy of $300$ keV. Secondly,we calculate contributions to the $S(300\,{\rm keV})$-factor from thesubthreshold $1^{-}$ and $2^{+}$ resonances, that are controlled by thesubthreshold ANCs. We also evaluate the contribution of the uncertainties ofthe subthreshold ANCs to the budget of the low-energy $S$-factor uncertainty,especially, the $S(300\,{\rm keV})$-factor. Thirdly, we analyse interference ofthe subthreshold resonances (SRs) with higher resonances and with the $E1$ and$E2$ direct captures to the ground state. Finally, we investigate a correlatedeffect of the subthreshold and ground-state ANCs on the low-energy $S$-factorand, in particular, on the $S(300\,{\rm keV})$-factor.

Asymptotic normalization coefficients for $\alpha+ {}^{12}{\rm C}$  synthesis and the $S$-factor for ${}^{12}{\rm C}(\alpha, \,\gamma){}^{16}{\rm  O}$ radiative capture