Can non-cyclic one-relator groups be finite? The answer appears to be no, but I can't find it anywhere. Worded another way, do there exist subgroups of finite index in the free group $F_2$ on two generators $x,y$ which is normally generated by a single element? It's clear that if the single element is a possible generator then the normal closure must be infinite index. However, what about elements which lie in characteristic subgroups?

The answer is no. A group defined by a presentation with more generators that relations is infinite, because its abelianization (i.e. the abelian group defined by this presentation) is infinite. This is just elementary linear algebra. See for example Finitely presented Group with less relations than Generators.

So if a presentation has more than one generator and just one relation then it defines an infite group.