Additive structure of polynomial rings

Question

I've been wondering recently about results for irreducibility that use the "additive structure" of the polynomial ring at hand. For instance, can we say anything about the irreducibility of a sum of two irreducible polynomials that satisfy some conditions? This seems like an additive number theory problem but in a different context. I searched the literature for stuff along these lines but did not find much. Am I missing something obvious (like maybe I can be asking the same questions in the underlying ring) or is this just a question with sparse literature around it? Any papers involving similar questions would be extremely appreciated!