  Recent discoveries of carbon-enhanced metal-poor stars like SMSS
J031300.36-670839.3 provide increasing observational insights into the
formation conditions of the first second-generation stars in the Universe,
reflecting the chemical conditions after the first supernova explosion. Here,
we present the first cosmological simulations with a detailed chemical network
including primordial species as well as C, C$^+$, O, O$^+$, Si, Si$^+$, and
Si$^{2+}$ following the formation of carbon-enhanced metal poor stars. The
presence of background UV flux delays the collapse from $z=21$ to $z=15$ and
cool the gas down to the CMB temperature for a metallicity of
Z/Z$_\odot$=10$^{-3}$. This can potentially lead to the formation of lower mass
stars. Overall, we find that the metals have a stronger effect on the collapse
than the radiation, yielding a comparable thermal structure for large
variations in the radiative background. We further find that radiative
backgrounds are not able to delay the collapse for Z/Z$_\odot$=10$^{-2}$ or a
carbon abundance as in SMSS J031300.36-670839.3.
