We have experimentally studied few-body impurity systems consisting of a single fermionic atom and a small bosonic field on the sites of an optical lattice. Quantum phase revival spectroscopy has allowed us to accurately measure the absolute strength of Bose-Fermi interactions as a function of the interspecies scattering length. Furthermore, we observe the modification of Bose-Bose interactions that is induced by the interacting fermion. Because of an interference between Bose-Bose and Bose-Fermi phase dynamics, we can infer the mean fermionic filling of the mixture and quantify its increase (decrease) when the lattice is loaded with attractive (repulsive) interspecies interactions.