Following a problem posed by Lovász in 1969, it is believed that every finite connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from finite groups having a (2,s,3) -presentation, that is, for groups G=\langle a,b\mid a^2=1, b^s=1, (ab)^3=1, \dots \rangle generated by an involution a and an element b of order s\geq3 such that their product ab has order 3 . More precisely, it is shown that the Cayley graph X=\operatorname{Cay}(G,\{a,b,b^{-1}\}) has a Hamilton cycle when |G| (and thus s ) is congruent to 2 modulo 4 , and has a long cycle missing only two adjacent vertices (and thus necessarily a Hamilton path) when |G| is congruent to 0 modulo 4 .