CONUS is a novel experiment aiming at detecting elastic neutrino nucleus scattering in the fully coherent regime using high-purity Germanium (Ge) detectors and a reactor as antineutrino ($bar u$) source. The detector setup is installed at the commercial nuclear power plant in Brokdorf, Germany, at a very small distance to the reactor core in order to guarantee a high flux of more than 10$^{13}bar u$/(s$cdot$cm$^2$). For the experiment, a good understanding of neutron-induced background events is required, as the neutron recoil signals can mimic the predicted neutrino interactions. Especially neutron-induced events correlated with the thermal power generation are troublesome for CONUS. On-site measurements revealed the presence of a thermal power correlated, highly thermalized neutron field with a fluence rate of (745$pm$30)cm$^{-2}$d$^{-1}$. These neutrons that are produced by nuclear fission inside the reactor core, are reduced by a factor of $sim$10$^{20}$ on their way to the CONUS shield. With a high-purity Ge detector without shield the $gamma$-ray background was examined including highly thermal power correlated $^{16}$N decay products as well as $gamma$-lines from neutron capture. Using the measured neutron spectrum as input, it was shown, with the help of Monte Carlo simulations, that the thermal power correlated field is successfully mitigated by the installed CONUS shield. The reactor-induced background contribution in the region of interest is exceeded by the expected signal by at least one order of magnitude assuming a realistic ionization quenching factor of 0.2.