  We begin by defining a type of K\"ahler metric near the zero section of a
trivial holomorphic open disk bundle $N$ over a compact K\"ahler manifold $X$
by incorporating flows generated by holomorphic vector fields on $X$. These
metrics are then shown to deviate exponentially from Poincar\'e-type metrics on
$N\setminus X$ in terms of the log-polar distance from $X$ in $N$. Lastly we
see that they arise naturally when perturbing classes containing
Poincar\'e-type K\"ahler metrics of constant scalar curvature to obtain nearby
cscK metrics even when the perturbed class on $X$ does not admit a cscK metric.
