The process requiring the absorption of energy
is

In option(A) $\mathrm{F}\to {\mathrm{F}}^{-}$

There is no need for absorption energy to add fluorine because it only needs one electron to achieve the noble gas configuration. So, it is an incorrect option.

In option(B) $\mathrm{Cl}\to {\mathrm{Cl}}^{-}$

It does not require absorption energy to add chlorine because it only needs one electron to achieve a noble gas configuration. So, it is also an incorrect option.

In option(C) ${\mathrm{O}}^{-}\to {\mathrm{O}}^{2-}$

When one electron is added to oxygen $\mathrm{O}:{\mathrm{s}}^{2}2{ \mathrm{s}}^{2}2{\mathrm{p}}^4$ it becomes ${\mathrm{O}}^{-}:1{ \mathrm{s}}^{2}2{ \mathrm{s}}^{2}2{\mathrm{p}}^5$

 Similarly, When another electron is added to oxygen $\left({\mathrm{O}}^{2-}:1{ \mathrm{s}}^{2}2{ \mathrm{s}}^{2}2{\mathrm{p}}^6\right)$, the electron that is already present repels the incoming electron. Hence, it needs more absorption energy to add an electron.

Therefore, option(C) is the correct answer.