The present paper investigates the impact of pairing correlation on the nuclear level density (NLD) and radiative strength function (RSF) of an odd-odd and rare-earth $^{166}$Ho nucleus. By employing the exact thermal pairing (EP) solution in conjunction with both the temperature-dependent independent-particle model (EP+IPM) for the NLD and the phonon damping model (EP+PDM) for the RSF, we achieve good agreement with the experimental NLD and RSF data, which have been recently extracted by using the Oslo method. Notably, without adding any extra parameters, our calculations successfully reproduce the pygmy dipole resonances (PDR) and a portion of the strength around the region of scissor resonance (SR) in the RSF. This suggests that the SR in this nucleus may be less pronounced than that obtained in the previous analysis within the phenomenological model. These features are the consequences of the EP and the couplings of the PDM phonon to all the particle-hole (ph), particle-particle (pp), and hole-hole (hh) configurations within the EP+PDM, thus insisting on the importance of the underlying physics involved in this model. Utilizing the obtained NLD and RSF, we predict the $^{165}$Ho${(n,\gamma )}^{166}$Ho and Maxwellian-averaged cross sections, which exhibit remarkable agreement with all the available experimental data from the Experimental Nuclear Reaction Data and Karlsruhe Astrophysical Database of Nucleosynthesis in Stars. These results reflect the imprint of the pairing correlation on those quantities.