Let denote the graph obtained from by adding all edges between the vertices of degree in it. We show how to adapt the argument of our previous paper [A.V. Kostochka, N. Prince, On -minors in graphs with given average degree, Discrete Math. 308 (2008) 4435–4445] to prove that if , then every graph with average degree at least has a minor. This refines a corresponding result by Kühn and Osthus.