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In supergravity, the dynamics of the sgoldstino -- superpartner of the goldstino superfield associated with the breaking of supersymmetry at low energy -- can substantially modify the dynamics of inflation in the primordial Universe. So-called sgoldstinoless models assume the existence of a nilpotency constraint $S^2=0$ that effectively removes the sgoldstino from the theory. Such models were proposed to realise non-oscillatory inflation scenarios with a single scalar field, which feature a long period of kination at the end of inflation, and therefore a non-standard post-inflationary cosmology. Using effective operators, we propose models in which the sgoldstino is stabilized close to the origin to reproduce the nilpotent constraint. We show that small sgoldstino fluctuations may lead to a sizeable back-reaction on the cosmological history. We study the effect of this back-reaction on the inflation observables measured in the cosmic microwave background and confront the model to a series of constraints including limits on $\Delta N_{\rm eff}$. We show that the peculiar form of the potential in the large supersymmetry breaking scale limit can generate peaks in the scalar power spectrum produced from inflation. We study how certain perturbation modes may re-enter the horizon during or after kination and show that a large supersymmetry breaking scale may lead to the formation of primordial black holes with various masses in the early Universe., Comment: 26 pages, 7 figures