  This paper is dedicated to $L^p$ bounds on eigenfunctions of a
Sch\"odinger-type operator $(-\Delta_g)^{\alpha/2} +V$ on closed Riemannian
manifolds for critically singular potentials $V$. The operator
$(-\Delta_g)^{\alpha/2}$ is defined spectrally in terms of the eigenfunctions
of $-\Delta_g$. We obtain also quasimodes and spectral clusters estimates. As
an application, we derive Strichartz estimates for the fractional wave equation
$(\partial_t^2+(-\Delta_g)^{\alpha/2}+V)u=0$. The wave kernel techniques
recently developed by Bourgain-Shao-Sogge-Yao and Shao-Yao play a key role in
this paper. We construct a new reproducing operator with several local
operators and some good error terms. Moreover, we shall prove that these local
operators satisfy certain variable coefficient versions of the "uniform Sobolev
estimates" by Kenig-Ruiz-Sogge. These enable us to handle the critically
singular potentials $V$ and prove the quasimode estimates.
