  We consider an extended theory of gravity with Lagrangian $\mathcal{L} =
f(R,{\bf T}^{(n)})$, with ${\bf T}^{(n)}$ being a $2n$-th order invariant made
of contractions of the energy-momentum tensor. When $n=1$ this theory reduces
to $f(R,T)$ gravity, where $T$ accounts for the trace of the energy-momentum
tensor. We study the gravitational wave polarization modes, from which it
results that when the matter Lagrangian contains dynamical scalar fields
minimally coupled to the geometry, further polarization modes arise with
respect to General Relativity. Finally we show that the motion for test
particles is non-geodesic and we explicitly obtain the extra-force.
