Dynamics of generalized Palatini theories of gravity

It is known that in $f(R)$ theories of gravity with an independent connection which can be both nonmetric and nonsymmetric, this connection can always be algebraically eliminated in favor of the metric and the matter fields, so long as it is not coupled to the matter explicitly. We show here that this is a special characteristic of $f(R)$ actions, and it is not true for actions that include other curvature invariants. This contradicts some recent claims in the literature. We clarify the reasons for this contradiction.