We study Galois covers of the projective line branched at three points with Galois group G. When such a cover is defined over a p-adic field, it is known to have potentially good reduction to characteristic p if p does not divide the order of G. We give a sufficient criterion for good reduction, even when p does divide the order of G, so long as the p-Sylow subgroup of G is cyclic and the absolute ramification index of a field of definition of the cover is small enough. This extends work of (and answers a question of) Raynaud. Our proof depends on working very explicitly with Kummer extensions of complete discrete valuation rings with imperfect residue fields.