  We consider a magnetic impurity coupled to both fermionic quasiparticles with
a pseudogap density of states and bosonic spin fluctuations. Using
renormalization group and large-N calculations we investigate the phase diagram
of the resulting Fermi-Bose Kondo model. We show that the Kondo temperature is
strongly reduced by low-energy spin fluctuations, and make connections to
experiments in cuprate superconductors. Furthermore we derive an exact exponent
for the critical behavior of the conduction electron T matrix, and propose our
findings to be relevant for certain scenarios of local quantum criticality in
heavy-fermion metals.
