  We generalize our previous theory [Nano Lett. \textbf{18}, 5015-5023 (2018)]
to investigate the influence of electron-phonon (e-p) coupling and vibrational
relaxation on photoinduced \textit{anomalous Coulomb blockade}, which
originates from the triplet states and the energy level alignment. We derive
the master equation for an irradiated molecular transistor and obtain the
relevant rates via the Redfield theory instead of the phenomenological Fermi
golden rule approach. To explore the interplay between e-p coupling and
vibrational relaxation, we analyze the charge stability diagrams and the
current-voltage characteristics (both gate voltage and source-drain bias
voltage) under different e-p coupling strengths in two extreme limits of
vibrational relaxation (equilibrated and unequilibrated phonon regimes). From
the perspective of energy level alignment, we choose four representative
situations and derive the analytical formulas of the photoinduced current in
the equilibrated regime. The analytical solution reveals a new type of
photocurrent due to e-p coupling that does not require the perfect energy level
alignment between charged states and triplet states. In general, our study
indicates that photoinduced current and anomalous Coulomb blockade caused by
the triplet states are supposed to be experimentally observed.
