  We study the nature of finite-time singularities for the Chern-Ricci flow,
partially answering a question of Tosatti-Weinkove. We show that a solution of
degenerate parabolic complex Monge-Amp\`ere equations starting from arbitrarily
positive (1,1)-currents are smooth outside some analytic subset, generalizing
works by Di Nezza-Lu. We extend Guedj-Lu's recent approach to establish uniform
a priori estimates for degenerate complex Monge-Amp\`ere equations on compact
Hermitian manifolds. We apply it to studying the Chern-Ricci flows on complex
log terminal varieties starting from an arbitrary current.
