Second ionization potential is the energy required to remove an electron from a uni-positively charged ion.
Let's look at the electronic configurations. 

$O$ $\to$ $2,6$  
$O^{+}$ $\to$ $2,5$

$F$ $\to$ $2,7$ 
$F^{+}$  $\to$ $2,6$

 $I.P_1$
$O$ $\to$ $O^{+}$   $F$ $\to$ $F^{+}$
 $I.P_2$

$O$ $\to$  $O^{2+}$  $F^{+}$ $\to$$F^{2+}$

$I.P_1$  of   $F>O$   as fluorine has smaller atomic size as compared to O.

$I.P_2$  of    $O>F$ as $O^{+}$ as it is harder to ionise the O+ compared to F+.