Is there a Lagrangian for Super-Gravity?

Consider the simple case of $mathcal{N}=1$ on-shell supergravity in 4 dimensions with no matter content. The full action is made up of the Einstein-Hilbert term and the Rarita-Schwinger action for the gravitino, with $$ S_{EH}=frac{1}{2kappa^2}intmathrm d^4 xdet e e^mu_ae^nu_bR^{ab}_{munu}(omega) $$ where $R_{munu}^{ab}$ is the field strength of the spin connection $omega_mu^{ab}$ and $det e$ takes the place of the volume element.

Since this SUGRA theory can formally be defined as a gauge theory with gauge group $text{SO}(1, d-1)$, the commutator of the supercovariant derivatives defines the Riemann tensor above: $$ [D_mu, D_nu]=frac14R_{munu}^{ab}gamma_{ab} $$

So $e^mu_ae^nu_bR^{ab}_{munu}(omega)$ is essentially the Ricci scalar in the spin-connection-vielbien formalism. The spin connection here is actually a function of the vielbien, and on introducing supersymmetry, also of the fermionic fields (which you can see when you find the EOM for $omega$ in the first-order formalism).

The presence of the scalar curvature in the action is a feature of all supergravity theories. There are of course Lagrangian descriptions for things like type IIA SUGRA as well, obtained as the low-energy effective action of type IIA string theory: $$ S = frac{1}{2kappa^2}intmathrm d^{10}xsqrt{-g}e^{-2phi}left(R+4partial_muPhipartial^muPhi-frac1{12}|H_3|^2right)-frac{1}{4kappa^2}intmathrm d^{10}xsqrt{-g}left(|F_2|^2+|tilde F_4|^2right)-frac{1}{4kappa^2}intmathrm d^{10}xsqrt{-g} B_2wedge F_4wedge F_4 $$ with the three terms corresponding respectively to the bosonic part, the RR part and the Chern-Simons part. You can remove the ugly $e^{-2phi}$ in the bosonic action by going to the Einstein frame through a conformal rescaling, whereupon you pick up the familiar Einstein-Hilbert term $intmathrm d^n x sqrt{-g} R$.