  In a recent paper, after introducing the notion of plus-one generated
hyperplane arrangements, Takuro Abe has shown that if we add (resp. delete) a
line to (resp. from) a free line arrangement, then the resulting line
arrangement is either free or plus-one generated. In this note we prove that
the same properties hold when we replace the line arrangement by a free curve
and add (resp. delete) a line. The proof uses a new version of a key result due
originally to H. Schenck, H. Terao and M. Yoshinaga, in which no quasi
homogeneity assumption is needed. Two conjectures about the Tjurina number of a
union of two plane curve singularities are also stated. As a geometric
application, we show that, under a mild numerical condition, the projective
closure of a contractible, irreducible affine plane curve is either free or
plus-one generated, using a deep result due to U. Walther.
