The following paper is relevant:
Finite coverings by normal subroups by Brodie, Chamberlain and Kapp, PAMS 1988.
The main focus is on coverings of infinite groups although their main theorem is still interesting in the finite case.
Theorem. A group has a nontrivial finite covering by normal subgroups if and only if it has a quotient isomorphic to an elementary abelian $p$-group of rank two for some prime $p$.
One of the corollaries to this theorem is also relevant for finite groups:
Corollary. Let $G=\bigcup\limits_{i=1}^n N_i$ where $N_1,\dots, N_n$ form an irredundant covering of $G$ by proper normal subgroups. Then $G/D$ with $D=\bigcap\limits_{i=1}^n N_i$ is finite and solvable.
This corollary effectively reduces the question of coverings by normal subgroups to the study of solvable groups. The paper can be viewed here. If you don't have access email me and I'll send it to you.
