Effective nuclear charge and Ionization energy
Effective nuclear charge $Z_{\text{eff}}$ is given by Slater's Rules.

According to these rules, for $\ce{C}$, effective nuclear charge experienced by an outer shell electron is:
$$Z_{\text{eff}} = 6 - 0.85\times 2 - 0.35\times 3 = 3.25$$

For $\ce{C+}$, effective nuclear charge is
$$Z_{\text{eff}} = 6 - 0.85\times 2 - 0.35\times 2=3.6$$

Since the effective charge for $\ce{C+}$ is more than that on $\ce{C}$, the second ionisation energy is more than that of the first.