We analyze intrinsic nonlinearities in two-dimensional (2D) polaritonic materials interacting with an optical wave. Focusing on the case of graphene, we show that the second-order nonlinear optical conductivity due to carrier density fluctuations associated with the excitation of a plasmon polariton is closely related to the ponderomotive force due to the oscillating optical field. A recent study (Sun et al 2018 Proc. Natl Acad. Sci. USA 115 3285-9) derived this force in the hydrodynamic regime of a generic Dirac fluid, and suggested that inclusion of interband transitions could have interesting implications. Here we reproduce the Drude-like result in a more general fashion on the basis of thermodynamics, which makes extension to other regimes straightforward. We find that for zero temperature a diverging nonlinearity is found at the interband threshold. By including finite-Temperature effects this is regularized, but remains quite significant even at room temperature. Going further beyond, we include nonlocal corrections as a second potential source of regularization, and find that they do not lead to broadening (as one would usually expect e.g. due to Landau damping), but rather to a splitting of the ponderomotive interband resonance, providing a very characteristic signature of nonlocality. Our analysis should prove useful to the open quest for exploiting nonlinearities in graphene and other 2D polaritonic materials, through effects such as photon drag.