  The frustrated magnetism on the Kondo lattice system motivates intriguing
Kondo-breakdown beyond the traditional Doniach scenario. Among them, the
fractionalized Fermi liquid (FL*) has drawn a particular interest by virtue of
its fractionalized nature. Here, we study the phase diagram of $J_{1}$-$J_{2}$
Kondo-Heisenberg model on a honeycomb lattice at a quarter filling. Employing
the slave-fermion mean-field theory with $d \pm id$ spin liquid ansatz and
exact diagonalization, we discuss the emergence of partial Kondo screening in
the frustrated regime with comparable $J_{1}$ and $J_{2}$, and the
fractionalized superconductor (SC*) which is superconductor analogy of the FL*.
Due to the larger number of local spin moments than itinerant electrons, the
magnetic fluctuation is still significant even in the strong-coupling limit,
which influences the thermodynamic and transport properties qualitatively. In
particular, we estimate the thermal conductance to probe the low-energy
excitation and show the anomalous behaviour in the SC* phase contrast to the
conventional superconductors.
