  Here we explore the role of temporal fluctuations in kinetic helicity on the
generation of large-scale magnetic fields in presence of a background linear
shear flow. Key techniques involved here are same as in our earlier work
\citep[][hereafter paper~I]{JS20}, where we have used the renovating flow based
model with shearing waves. Both, the velocity and the helicity fields, are
treated as stochastic variables with finite correlation times, $\tau$ and
$\tau_h$, respectively. Growing solutions are obtained when $\tau_h > \tau$,
even when this time-scale separation, characterised by $m=\tau_h/\tau$, remains
below the threshold for causing the turbulent diffusion to turn negative. In
regimes when turbulent diffusion remains positive, and $\tau$ is on the order
of eddy turnover time $T$, the axisymmetric modes display non-monotonic
behaviour with shear rate $S$: both, the growth rate $\gamma$ and the
wavenumber $k_\ast$ corresponding to the fastest growing mode, first increase,
reach a maximum and then decrease with $|S|$, with $k_\ast$ being always
smaller than eddy-wavenumber, thus boosting growth of magnetic fields at large
length scales. The cycle period $P_{\rm cyc}$ of growing dynamo wave is
inversely proportional to $|S|$ at small shear, exactly as in the fixed kinetic
helicity case of paper~I. This dependence becomes shallower at larger shear.
Interestingly enough, various curves corresponding to different choices of $m$
collapse on top of each other in a plot of $m P_{\rm cyc}$ with $|S|$.
