  We prove that, given a compact Riemann surface $\Sigma$ and disjoint finite
sets $\varnothing\neq E\subset\Sigma$ and $\Lambda\subset\Sigma$, every map
$\Lambda \to \mathbb{R}^3$ extends to a complete conformal minimal immersion
$\Sigma\setminus E\to \mathbb{R}^3$ with finite total curvature.
  This result opens the door to study optimal hitting problems in the framework
of complete minimal surfaces in $\mathbb{R}^3$ with finite total curvature. To
this respect we provide, for each integer $r\ge 1$, a set
$A\subset\mathbb{R}^3$ consisting of $12r+3$ points in an affine plane such
that if $A$ is contained in a complete nonflat orientable immersed minimal
surface $X\colon M\to\mathbb{R}^3$, then the absolute value of the total
curvature of $X$ is greater than $4\pi r$.
