There's an extensive literature on this and more general questions.
First, let's consider a more precise formulation of the question.
Let $H$ be a handlebody, and let $K\subset H$ be a knot. Let's
assume that $\partial H \subset H-K$ is incompressible. Otherwise,
$H=H_0 \natural H_1$ the boundary connect sum of two handlebodies,
such that $K\subset H_0$, and $\partial H_0 \subset H_0-K$ is
incompressible. Then a surgery on $K$ makes $H$ a handlebody if
and only if the corresponding surgery makes $H_0$ a handlebody.
Thus, one may reduce to considering the case that $K$ is diskbusting.
With this reformulation of the question, the case that $K$ is
a core curve corresponds to when $H$ is a solid torus, and $K$ is
a core curve (since this is the only case in which a core curve is
diskbusting). So assume that $K$ is not a core curve. If $K$
is isotopic into $\partial H$, then there are infinitely many surgeries
which yield $H$. In this case, there is an annulus going between the
knot and $\partial H$. The surgery slopes which intersect the annulus
slope once give back $H$. Surgery along the annular slope gives
a manifold containing an incompressible surface, by Jaco's lemma.
If the intersection with the annulus slope is $>1$, then the manifold
has incompressible boundary by Theorem 2.4.3 of the cyclic surgery paper.
A result of Wu implies that for knots $K$ which
are not isotopic into the boundary, the distance between boundary-reducible
surgeries is at most one, and therefore there are at most two non-trivial
surgeries which may yield a handlebody.
In the case that $H$ is a solid torus, it was proved by Berge
and Gabai that $K$ must be a 1-bridge braid, and a complete description
was given. There is a famous example (the Berge link) which yields 3 solid torus surgeries,
which shows that Wu's estimate is sharp (there is a 3-fold symmetry permuting
the slopes).
Wu has further results on the case of 1-bridge knots in handlebodies.
Added: Frigerio, Martelli and Petronio show that there are 1-bridge
knots in handlebodies with three handlebody fillings (including the trivial one),
generalizing the example of the Berge link to higher genus boundary, and showing
that Wu's theorem is sharp in general. It seems to be an open question whether
all such examples are 1-bridge (see discussion in another of Wu's papers).
