Exceptional sets related to Hayman's alternative

Let E be a subset of the complex plane C consisting of a countable set of points tending to ∞ and let k⩾1 be an integer. We derive a spacing condition (dependent on k) on the points of E which ensures that, if f is a function meromorphic in C with sufficiently large Nevanlinna deficiency at the poles, then either f takes every complex value infinitely often, or the kth derivative f(k) takes every non-zero complex value infinitely often, in C−E. This improves a previous result of Langley.