  Astrochemistry has been widely developed as a power tool to probe physical
properties of the interstellar medium (ISM) in various conditions of the Milky
Way (MW) Galaxy, and in near and distant galaxies. Most current studies
conventionally apply linear scaling to all elemental abundances based on the
gas-phase metallicity. However, these elements, including carbon and oxygen,
are enriched differentially by stellar nucleosynthesis and the overall galactic
chemical evolution, evident from $\alpha$-enhancement in multiple galactic
observations such as starbursts, high-redshift star-forming galaxies, and
low-metallicity dwarfs. We perform astrochemical modeling to simulate the
impact of an $\alpha$-enhanced ISM gas cloud on the abundances of the three
phases of carbon (C$^+$, C, CO) dubbed as `the carbon cycle'. The ISM
environmental parameters considered include two cosmic-ray ionization rates
($\zeta_{\rm CR}=10^{-17}$ and $10^{-15}\,{\rm s}^{-1}$), two isotropic FUV
radiation field strengths ($\chi/\chi_0=1$ and $10^2$), and (sub-)linear
dust-to-gas relations against metallicity, mimicking the ISM conditions of
different galaxy types. In galaxies with [C/O] $<$ 0, CO, C and C$^+$ all
decrease in both abundances and emission, though with differential biases. The
low-$J$ CO emission is found to be the most stable tracer for the molecular
gas, while C and C$^+$ trace H$_2$ gas only under limited conditions, in line
with recent discoveries of [CI]-dark galaxies. We call for caution when using
[CII]~$158\mu$m and [CI](1-0) as alternative H$_2$-gas tracers for both diffuse
and dense gas with non-zero [C/O] ratios.
