  In present paper we propose further modification of $f(R,T)$-gravity (where
$T$ is trace of energy-momentum tensor) by introducing higher derivatives
matter fields. We discuss stability conditions in proposed theory and find
restrictions for parameters to prevent appearance of main type of
instabilities, such as ghost-like and tachyon-like instabilities. We derive
cosmological equations for a few representations of theory and discuss main
differences with convenient $f(R,T)$-gravity without higher derivatives. It is
demonstrated that in presented theory inflationary scenarios appears quite
naturally even in the dust-filled Universe without any additional matter
sources. Finally we construct inflationary model in one of the simplest
representation of the theory, calculate main inflationary parameters and find
that it may be in quite agreement with observations.
