D

$\because |x|=5$，$|y|=9$，
$\therefore x=-5$或$5$，$y=-9$或$9$，
$\because x-y \gt 0$，
$\therefore x \gt y$，
$\therefore y$不能是$9$，
$\therefore x=5$，$y=-9$或$x=-5$，$y=-9$，
$\therefore x+y=5+\left(-9\right)=-4$或$x+y=-5+\left(-9\right)=-14$，
$\therefore x+y$的值是$-4$或$-14$.
故选：$D$.