  Within the framework of the modified potential cluster model with forbidden
states, the $^{13}$N($p,\gamma $)$^{14}$O reaction rate and the astrophysical
$S$-factor are considered. It is shown that the first $p^{13}$% N resonance
determines the $S$-factor and contributions of the $M1$ and $E2$ transitions
are negligible at energies $E<1$ MeV, but are significant at high energies. The
$S$-factor strongly depends on the $^{3}S_{1}$ resonance parameters. The
influence of the width of \ the $^{3}S_{1}$ resonance on $S$% -factor is
demonstrated. The reaction rate is calculated and an analytical approximation
for the reaction rate is proposed. A comparison of our calculation with
existing data is addressed. Results of our calculations for the
$^{13}$N($p,\gamma )^{14}$O reaction rate provide the contribution to the
steadily improving reaction rate database libraries. Our calculations of the $%
^{13}$N($p,\gamma )^{14}$O reaction rate along with results for the rates of
$^{14}$N($% p,\gamma )^{15}$O and $^{12}$C$(p,\gamma )^{13}$N processes provide
the temperature range $0.13<T_{9}<0.97$ for the conversion of CNO cycle to the
HCNO cycle. Our results demonstrate that at early stages of a nova explosion at
temperatures about $0.1$ $T_{9}$ and at late stages of evolution of
supermassive stars at temperatures about $1.0$ $T_{9}$ the ignition of the HCNO
cycle could occur at much lower densities of a stellar medium.
