  In this paper, we prove a Mergelyan type approximation theorem for immersed
holomorphic Legendrian curves in an arbitrary complex contact manifold
$(X,\xi)$. Explicitly, we show that if $S$ is a compact admissible set in a
Riemann surface $M$ and $f:S\to X$ is a $\xi$-Legendrian immersion of class
$\mathscr{C}^{r+2}(S,X)$ for some $r\ge 2$ which is holomorphic in the interior
of $S$, then $f$ can be approximated in the $\mathscr{C}^r(S,X)$ topology by
holomorphic Legendrian embeddings from open neighbourhoods of $S$ into $X$.
This result has numerous applications, some of which are indicated in the
paper. In particular, by using Bryant's correspondence for the Penrose twistor
map $\mathbb{CP}^3\to S^4$ we show that a Mergelyan approximation theorem and
the Calabi-Yau property hold for superminimal surfaces in the $4$-sphere $S^4$.
