We consider corrections to the Einstein-Hilbert action which contain both higher order and nonlocal terms. We derive an effective Newtonian gravitational constant applicable at the weak field limit and use the primordial nucleosynthesis (BBN) bound and the local gravity constraints on $G_{eff}$ in order to test the viability of several cases of our general Lagrangian. We will also provide a BBN constrain on the $\Box R$ gravitational correction.