Summary: A circular-arc graph $G$ is the intersection graph of a collection of arcs on the circle and such a collection is called a model of $G$. Say that the model is proper when no arc of the collection contains another one, it is Helly when the arcs satisfy the Helly Property, while the model is proper Helly when it is simultaneously proper and Helly. A graph admitting a Helly (resp. proper Helly) model is called a Helly (resp. proper Helly) circular-arc graph. The clique graph $K(G)$ of a graph $G$ is the intersection graph of its cliques. The iterated clique graph $K^i(G)$ of $G$ is defined by $K^{0}(G)=G$ and $K^{i+1}(G) = K(K^i(G))$. In this paper, we consider two problems on clique graphs of circular-arc graphs. The first is to characterize clique graphs of Helly circular-arc graphs and proper Helly circular-arc graphs. The second is to characterize the graph to which a general circular-arc graph $K$-converges, if it is $K$-convergent. We propose complete solutions to both problems, extending the partial results known so far. The methods lead to linear time recognition algorithms, for both problems.