Step 1: Understanding second ionisation potential.
Second ionisation potential refers to the energy required to remove an electron from a singly charged positive ion. The stability of the resulting ion plays a crucial role.
Step 2: Electronic configurations after first ionisation.
C$^+$ : $1s^2\,2s^2\,2p^1$
N$^+$ : $1s^2\,2s^2\,2p^2$
O$^+$ : $1s^2\,2s^2\,2p^3$ (half-filled, extra stable)
F$^+$ : $1s^2\,2s^2\,2p^4$
Step 3: Comparing stability.
O$^+$ has a half-filled $2p^3$ configuration, making removal of the next electron difficult. Hence, oxygen has the highest second ionisation potential among the given elements.
Step 4: Conclusion.
Therefore, the correct increasing order of second ionisation potential is:
C < N < F < O