Is the union of a non-regular and a regular language regular?
There is no simple rule.

Consider the languages


$L_1 = \{ a^n b^n \mid n \geq 0\}$,
$L_2 = \{ b^n \mid n \geq 0 \}$, and
$L_3 = \{ a^n b^m \mid n,m \geq 0 \}$.


Clearly, $L_2$ and $L_3$ are regular while $L_1$ is not. Furthermore,


$L_1 \cup L_2$ is not regular but
$L_1 \cup L_3$ is.


Therefore, you will have to look more closely at the language at hand and maybe try to proof both claims. Our reference material can help you get started; feel free to post any specific questions about your attempts as new questions.