  In this work, we investigate the time evolution of the pseudo-(R\'enyi)
entropy after local primary operator quenches in 2D CFTs with $T\bar T/J\bar
T$-deformation. Using perturbation theory, we analyze the corrections to the
second pseudo-R\'enyi entropy at the late time, which exhibit a universal form,
while its early-time behavior is model-dependent. Moreover, we uncover
nontrivial time-dependent effects arising from the first-order deformation of
the $k^{\rm th}$ pseudo-R\'enyi entropy at the late time. Additionally, drawing
inspiration from the gravitational side, specifically the gluing of two cutoff
AdS geometries, we investigate the $k^{\rm th}$ pseudo-R\'enyi entropy for
vacuum states characterized by distinct $T\bar{T}$-deformation parameters, as
well as for primary states acting on different deformed vacuum states. Our
findings reveal additional corrections compared to the results of
pseudo-R\'enyi entropy for globally deformed vacuum states.
