Verbal sets and cyclic coverings

We consider groups G such that the set of all values of a fixed word w in G is covered by a finite set of cyclic subgroups. Fernández-Alcober and Shumyatsky studied such groups in the case when w is the word [x1,x2], and proved that in this case the corresponding verbal subgroup G′ is either cyclic or finite. Answering a question asked by them, we show that this is far from being the general rule. However, we prove a weaker form of their result in the case when w is either a lower commutator word or a non-commutator word, showing that in the given hypothesis the verbal subgroup w(G) must be finite-by-cyclic. Even this weaker conclusion is not universally valid: it fails for verbose words.