In our preceding paper we have introduced the notion of an s-homogeneous triple. In this paper we use this technique to study connected s-homogeneous algebras with two relations. For such algebras, we describe all possible pairs (A,M), where A is the s-Veronese ring and M is the (s,1)-Veronese bimodule of the s-homogeneous dual algebra. For each such a pair we classify all the corresponding algebras or give a necessary and sufficient conditions for the algebra under consideration correspond to a given pair. During this process, we have proved that an s-homogeneous algebra with two dimensional s-th component cannot be s-Koszul for s>2 and discuss the Koszul property for s-homogeneous algebras with two relations.