  We prove that if $f$ is a polynomial over a number field $K$ with a finite
superattracting periodic point and a non-archimedean place of bad reduction,
then there is an $\epsilon>0$ such that only finitely many $P\in K^{\text{ab}}$
have canonical height less than $\epsilon$ with respect to $f$. The key
ingredient is the geometry of the filled Julia set at a place of bad reduction.
We also prove a conditional uniform boundedness result for the $K$-rational
preperiodic points of such polynomials, as well as a uniform lower bound on the
canonical height of non-preperiodic points in $K$. We further prove
unconditional analogues of these results in the function field setting.
