  We obtain a priori local pointwise second derivative estimates for solutions
$u$ to a class of augmented Hessian equations on Riemannian manifolds, in terms
of the $C^1$ norm and certain $W^{2,p}$ norms of $u$. We consider the case that
no structural assumptions are imposed on either the augmenting term or the
right hand side of the equation, and the case where these terms are convex in
the gradient variable. In the latter case, under an additional ellipticity
condition we prove that the dependence on any $W^{2,p}$ norm can be dropped.
Our results are derived using integral estimates.
