  We consider scalar field inflation in the Palatini formulation of general
relativity. The covariant derivative of the metric is then non-zero. From the
effective theory point of view it should couple to other fields. We write down
the most general couplings between it and a scalar field that are quadratic in
derivatives. We consider both the case when the torsion is determined by the
field equations and the case when it is assumed to be zero a priori. We find
the 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 of
the 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 with
a large tensor-to-scalar ratio, even if the non-minimal coupling is zero.
