  In this paper, we first study infinitesimal deformations of a Lie conformal
algebra and a Lie conformal algebra with a module (called an $\mathsf{LCMod}$
pair), which lead to the notions of Nijenhuis operator on the Lie conformal
algebra and Nijenhuis structure on the $\mathsf{LCMod}$ pair, respectively.
Then by adding compatibility conditions between Nijenhuis structures and
$\mathcal{O}$-operators, we introduce the notion of an $\mathcal{O}
N$-structure on an $\mathsf{LCMod}$ pair and show that an $\mathcal{O}
N$-structure gives rise to a hierarchy of pairwise compatible
$\mathcal{O}$-operators. In particular, we show that compatible
$\mathcal{O}$-operators on a Lie conformal algebra can be characterized by
Nijenhuis operators on Lie conformal algebras. Finally, we introduce the
notions of conformal $r$-matrix-Nijenhuis structure and symplectic-Nijenhuis
structure on the Lie conformal algebra and study their relations.
