  Let $M$ be an open Riemann surface and $A$ be the punctured cone in
$\mathbb{C}^n\setminus\{0\}$ on a smooth projective variety $Y$ in
$\mathbb{P}^{n-1}$. Recently, Runge approximation theorems with interpolation
for holomorphic immersions $M\to\mathbb{C}^n$, directed by $A$, have been
proved under the assumption that $A$ is an Oka manifold. We prove analogous
results in the algebraic setting, for regular immersions directed by $A$ from a
smooth affine curve $M$ into $\mathbb{C}^n$. The Oka property is naturally
replaced by the stronger assumption that $A$ is algebraically elliptic, which
it is if $Y$ is uniformly rational. Under this assumption, a homotopy-theoretic
necessary and sufficient condition for approximation and interpolation emerges.
We show that this condition is satisfied in many cases of interest.
