The answer is no if $k$ is independent on $\lambda$. Take a $2$-dimensional torus with a flat metric, which is a very nice manifold. Eigenfunctions can be found explicitly, and they are analytic. There can be some points where many nodal lines intersect. Near these points, eigenfunctions will look like
$u(z)=\Re z^n$ where $n$ can be large. So for every $k$ we can we can find $n$
such that the integral of $1/u^k$ diverges. But it looks plausible that the integral will converge with some $k$ depending on $\lambda$ (and on the manifold).