We study functional identities of degreem. As an application of obtained results, we describe additive mappingsfi:A→Aof a prime ring A satisfying a functional identity of the formProvided thatAis not algebraic of bounded degree ≤nover the extended centroid. A number of results onk-commuting additive mappings and commuting traces ofm-additive mappings, obtained earlier, are special cases of our ones.