  We prove that if $B$ is a $k$-positive holomorphic line bundle on a compact
hyperk\"ahler manifold $M,$ then $H^p (M,\Omega^q\otimes B)=0$ for
$p>n+[\frac{k}{2}]$ and any nonnegative integer $q.$ In a special case $k=0$
and $q=0$ we recover a vanishing theorem of Verbitsky's with a little stronger
assumption.
