As it is well known, a balanced (and in particular ribbon) monoidal category is an algebra over the framed little 2-discs operad. The latter is homotopy equivalent to the operad of moduli space of genus 0 surfaces with boundaries, hence has a natural cyclic structure.

What additional structure correspond to a cyclic algebra in categories over this operad ?

Usually, in vector spaces, a cyclic algebra over a cyclic operad has a non-degenerate pairing compatible with the operadic operations. One marvelous property of categories is that the already have a canonical pairing given by hom spaces, and under mild conditions it is non-degenerate. So I guess that a cyclic structure in that case should be a certain compatibility between the twist/ribbon element and the home spaces.