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.