  We study the problem of finding a minimum weight connected subgraph spanning
at least $k$ vertices on planar, node-weighted graphs. We give a
$(4+\eps)$-approximation algorithm for this problem. We achieve this by
utilizing the recent LMP primal-dual $3$-approximation for the node-weighted
prize-collecting Steiner tree problem by Byrka et al (SWAT'16) and adopting an
approach by Chudak et al. (Math.\ Prog.\ '04) regarding Lagrangian relaxation
for the edge-weighted variant. In particular, we improve the procedure of
picking additional vertices (tree merging procedure) given by Sadeghian (2013)
by taking a constant number of recursive steps and utilizing the limited
guessing procedure of Arora and Karakostas (Math.\ Prog.\ '06). More generally,
our approach readily gives a $(\nicefrac{4}{3}\cdot r+\eps)$-approximation on
any graph class where the algorithm of Byrka et al.\ for the prize-collecting
version gives an $r$-approximation. We argue that this can be interpreted as a
generalization of an analogous result by K\"onemann et al. (Algorithmica~'11)
for partial cover problems. Together with a lower bound construction by Mestre
(STACS'08) for partial cover this implies that our bound is essentially best
possible among algorithms that utilize an LMP algorithm for the Lagrangian
relaxation as a black box. In addition to that, we argue by a more involved
lower bound construction that even using the LMP algorithm by Byrka et al.\ in
a \emph{non-black-box} fashion could not beat the factor $\nicefrac{4}{3}\cdot
r$ when the tree merging step relies only on the solutions output by the LMP
algorithm.
