Is the symmetric difference of a non regular language L and a finite language f non regular?
Denoting symmetric difference by $\Delta$, note that $(L \Delta f) \Delta f = L$. This suggests the following strategy: show that the symmetric difference of a regular language and a finite language is regular. This implies your claim via proof by contradiction.

The product construction shows the stronger fact that the symmetric difference of two regular languages is regular. Hence you can strengthen your claim by replacing "finite" with "regular".