Constant cycle curves on a K3 surface $X$ over $\mathbb{C}$ have been introduced by Huybrechts (2014) as curves whose points all define the same class in the Chow group. In this paper we study correspondences $Z \subseteq X\times X$ over $\mathbb{C}$ acting on the group $\mbox{ccc}(X)$ of cycles generated by irreducible constant cycle curves. We construct for any $n\geq 2$ and any very ample line bundle $L$ a locus $Z_n(L)\subseteq X\times X$ of expected dimension $2$, which yields a correspondence that acts on $\mbox{ccc}(X)$, when it has the expected dimension. We provide examples of $Z_n(L)$ for low $n$ and exhibit one correspondence different from $Z_n(L)$ acting on $\mbox{ccc}(X)$.