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We consider scalar field inflation in the Palatini formulation of generalrelativity. The covariant derivative of the metric is then non-zero. From theeffective theory point of view it should couple to other fields. We write downthe most general couplings between it and a scalar field that are quadratic inderivatives. We consider both the case when the torsion is determined by thefield equations and the case when it is assumed to be zero a priori. We findthe metric derivative terms can significantly modify inflationary predictions.We specialise to Higgs inflation and terms of only up to dimension 4.Transforming to the Einstein frame, we show that by tuning the coefficients ofthe new terms, we can generate various effective inflationary potentials,including quadratic, hilltop-type, $\alpha$-attractor and inflection point.Some of these can give inflation in agreement with observations, including witha large tensor-to-scalar ratio, even if the non-minimal coupling is zero.

Higgs inflation in the Palatini formulation with kinetic terms for the metric