  We present a theory of large-scale dynamo action in a turbulent flow that has
stochastic, zero-mean fluctuations of the $\alpha$ parameter. Particularly
interesting is the possibility of the growth of the mean magnetic field due to
Moffatt drift, which is expected to be finite in a statistically anisotropic
turbulence. We extend the Kraichnan-Moffatt model to explore effects of finite
memory of $\alpha$ fluctuations, in a spirit similar to that of Sridhar & Singh
(2014), hereafter SS14. Using the first-order smoothing approximation, we
derive a linear integro-differential equation governing the dynamics of the
large-scale magnetic field, which is non-perturbative in the
$\alpha$-correlation time $\tau_{\alpha}$. We recover earlier results in the
exactly solvable white-noise (WN) limit where the Moffatt drift does not
contribute to the dynamo growth/decay. To study finite memory effects, we
reduce the integro-differential equation to a partial differential equation by
assuming that the $\tau_{\alpha}$ be small but nonzero and the large-scale
magnetic field is slowly varying. We derive the dispersion relation and provide
explicit expression for the growth rate as a function of four independent
parameters. When $\tau_{\alpha}\neq 0$, we find that: (i) in the absence of the
Moffatt drift, but with finite Kraichnan diffusivity, only strong
$\alpha$-fluctuations can enable a mean-field dynamo; (ii) in the general case
when also the Moffatt drift is nonzero, both, weak or strong $\alpha$
fluctuations, can lead to a large-scale dynamo; and (iii) there always exists a
wavenumber ($k$) cutoff at some large $k$ beyond which the growth rate turns
negative. Thus we show that a finite Moffatt drift can always facilitate
large-scale dynamo action if sufficiently strong, even in case of weak $\alpha$
fluctuations, and the maximum growth occurs at intermediate wavenumbers.
