Finite corrections to the Higgs mass in SUSY

Question

I am worried here about specific supersymmetric scenarios in which an energy scale above the electroweak scale is explicitly introduced. An example would be the Seesaw extension of the Minimal Supersymmetric Standard Model (MSSM). My questions is:

In such scenarios, and even though SUSY guarantees a cancellation of the quadratic divergences afflicting the Higgs mass, isn't it true that finite corrections to the Higgs mass appear nonetheless - for instance proportional to the Majorana mass in the Seesaw example? Can SUSY cancel those as well?

More explicitly, do nonrenormalization theorems also guarantee that all finite corrections of the effective action vanish?