<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mmultiscripts> <mml:mi mathvariant="normal">N</mml:mi> <mml:mprescripts/> <mml:none/> <mml:mn>14</mml:mn> </mml:mmultiscripts> <mml:mo>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</

In stellar hydrogen burning, the CNO cycle dominates, with the $^{14}$N(p,$γ)^{15}$O reaction being the slowest process. Consequently, this reaction critically influences the solar composition, CNO neutrino fluxes, and the evolution of star clusters and galaxies. Recent direct measurements of $^{14}$N(p,$γ)^{15}$O have reported an enhanced astrophysical $S$-factor. This work presents a microscopic theoretical study of the $^{14}$N(p,$γ)^{15}$O reaction using the Gamow shell model in the coupled-channel representation (GSM-CC). The calculations achieve good agreement with experimental data for both the total $S$-factors and the separate contributions from transitions to the ground state and excited states of $^{15}\mathrm{O}$. However, the predicted $S$-factor at zero energy exceeds the experimental value. Based on the computed $S$-factors, the derived carbon and nitrogen abundances align closely with predictions from recent $^{14}$N(p,$γ)^{15}$O cross-section measurements, yet remain significantly lower than the latest solar neutrino observation values.