Unconventional quantum criticality in heavy-fermion systems has been extensively analyzed in terms of a critical destruction of the Kondo effect. Motivated by a recent demonstration of quantum criticality in a mixed-valent heavy-fermion system, -YbAlB4, we study a particle-hole-asymmetric Anderson impurity model with a pseudogapped density of states. We demonstrate Kondo destruction at a mixedvalent quantum critical point, where a collapsing Kondo energy scale is accompanied by a singular charge-fluctuation spectrum. Both spin and charge responses scale with energy over temperature (!=T) and magnetic field over temperature (H=T). Implications for unconventional quantum criticality in mixedvalence heavy fermions are discussed.