Boundary terms for stringy correction to GR We know that there can be possible higher derivative corrections (stringy corrections) to the Einstein-Hilbert action. In GR, to ensure that we get the Einstein Field equations from varying the E-H action, we necessarily have to add the Gibbons-Hawking-York boundary term (Wikipedia link to GHY), as well as a counterterm (to ensure the action is finite). 
Let's say we consider adding higher derivative corrections (also known as stringy corrections) to the Einstein-Hilbert action, i.e we include the Gauss-Bonnet term, as well as the (Riemann)^3 term and so on and so forth.
However, would we need further boundary terms, and perhaps counterterms too beyond the GHY term to ensure that we obtain the correct E.O.M from $\delta S=0$. 
To summarize my question: What is the nature of these boundary terms (beyond the GHY term)? 
 A: It really depends on how you interpret these higher order terms. For example, in the case of treating it as a LEEFT, one can simply just add these terms on and the perform perturbative calculations in the low energy limit. In this case we find that at low energies we retain GR results, with the higher derivative terms contributing via a cut-off mass dependent coupling. For example, in Einstein-Gauss-Bonnet-Scalar effective field theory, one has the action,
$$
S\propto \int\sqrt{-g}(M_P^2R+\partial_\mu\phi\partial^\mu\phi~+\frac{\alpha}{\Lambda}\phi\mathcal{G}+\frac{c_1}{\Lambda^2}R^3+\frac{c_2}{\Lambda^2}R_{\mu\nu}\square R^{\mu\nu}+...),
$$
and we can go away and calculate a whole load of stuff, provided we treat the theory classically.
The Quantum case is a little more complex, because we would have to add counter terms on in order to keep loop amplitudes finite. However in terms of the consequences of these counter terms and boundary terms, they will be suppressed by a cut-off mass $\Lambda$ as above. It is perfectly reasonable to add these stringy corrections to the theory, and the counter terms and boundary terms that come along with all of that but due to the suppression of such terms (in order for the dimensions of each operator to be correct in the action) they will not contribute at the energies where GR is an accurate theory. For an example of this kind of analysis I would recommend looking at The Speed of Gravity. 
I should stress that when I say low energy, that is quite ambiguous and depends on the LEEFT you're looking at. For a bottom-up EFT (one where we begin with GR and add the corrections) this "low energy limit" can actually be quite high energies. I think the latest tests of GR show it to be accurate to energies around 1$mm$ in length. 
Now all of the above applies to a manifold without a boundary. With a boundary one would still have to suppress the operators by some mass scale in order to have the correct units, however now the BTs will contribute to the dynamics of the system. So as long as you look at this as an EFT problem, a lot of the issues are taken care of by this suppression mass. However, the connection between String theory, LEEFTs and GR is still an area that people are exploring and with better EFT techniques we will hopefully be able to treat the theories with more rigour. 
If I haven't made anything clear, or the OP wants a little more detail please just let me know. 
