  We expand the existing polaron response theory, expressed within the
Mori-Zwanzig projection operator formalism applicable for the transfer of
arbitrary energy and zero momentum, to the case of finite momentum exchange. A
general formula is then derived which can be used to calculate the response of
a system to a probe that transfers both momentum and energy to the system. The
main extension of the existing polaron response theory is the finite momentum
exchange which was not needed until now since it is negligible for optical
absorption. However, this formalism is needed to calculate the response of the
polaronic system consisting of an impurity in a Bose-Einstein condensate to
Bragg spectroscopy. We show that the well-known features that appear in the
optical absorption of the solid state Fr\"ohlich polaron are also present in
the Bragg response of the BEC-impurity polaron. The f-sum rule is written in a
form suitable to provide an independent consistency test for our results. The
effect of lifetime broadening on the BEC-impurity spectrum is examined. The
results derived here are discussed in the framework of an experimental
realization consisting of a lithium impurity in a sodium condensate.
