Let $G$ be a connected simply-connected reductive algebraic group. In this article, we consider the normal algebraic varieties equipped with a horospherical $G$-action such that the quotient of a $G$-stable open subset is a curve. Let $X$ be such a $G$-variety. Using the combinatorial description of Timashev, we describe the class group of $X$ by generators and relations and we give a representative of the canonical class. Moreover, we obtain a smoothness criterion for $X$ and a criterion to determine whether the singularities of $X$ are rational or log-terminal respectively.