A d-relaxed k-edge colouring is an edge colouring using colours from the set {1,. .. , k} such that each monochromatic set of edges induces a subgraph with maximum degree at most d. A neighbour sum distinguishing d-relaxed k-edge colouring of G is a d-relaxed k-edge colouring such that for each edge uv ∈ E(G), the sum of colours taken on the edges incident to u is different from the sum of colours taken on the edges incident to v. By χ d (G) we denote the smallest value k in such a colouring of G. In this paper, we prove that χ 2 (G) ≤ 4 for every connected subcubic graph with at least three vertices. For complete graphs with at least three vertices, we show that χ d (K n) ≤ 4 if d ∈ { (n − 1)/2 ,. .. , n − 1} and we also determine the exact value of χ 2 (K n). Finally, we determine the value of χ d (T) for any tree T with at least three vertices.