  We consider models of a scalar field coupled to quadratic $R\!+\!R^2$ gravity
in the framework of the Palatini formulation. The resulting Einstein-frame
generalized $k$-inflation effective theory is analyzed assuming that the
constant-roll condition holds. We focus on a quartic self-interaction
potential, a case of particular appeal modelling Higgs inflation, considering
the cases of minimal and non-minimal coupling of the inflaton to gravity. For
an appropriate range of the model parameters in the large field domain the
obtained values for the inflationary observables are found in agreement with
current observations.
