Let A be a primitive permutation group and G a normal subgroup of A such that A/G is cyclic. Let a be a generator for A/G. Motivated by questions arising in connection to coverings of smooth connected projective curves, we study the proportion of derangements in the coset aG. We use the Aschbacher-O'Nan-Scott theorem for primitive groups to partition the problem and provide answers in the affine case.