  Andreae (1986) proved that the cop number of connected $H$-minor-free graphs
is bounded for every graph $H$. In particular, the cop number is at most
$|E(H-h)|$ if $H-h$ contains no isolated vertex. The main result of this paper
is an improvement on this bound, which is most significant when $H$ is small or
sparse, for instance when $H-h$ can be obtained from another graph by multiple
edge subdivisions. Some consequences of this result are improvements on the
upper bound for the cop number of $K_{3,m}$-minor-free graphs, $K_{2,m}$-minor
free graphs and linklessly embeddable graphs.
