  In this paper, we first use the distribution of the number of records to
demonstrate that the right tail probabilities of counts of rare events are
generally better approximated by the right tail probabilities of Poisson
distribution than {those} of normal distribution. We then show the moderate
deviations in Poisson approximation generally require an adjustment and, with
suitable adjustment, we establish better error estimates of the moderate
deviations in Poisson approximation than those in \cite{CFS}. Our estimates
contain no unspecified constants and are easy to apply. We illustrate the use
of the theorems in six applications: Poisson-binomial distribution, matching
problem, occupancy problem, birthday problem, random graphs and 2-runs. The
paper complements the works of \cite{CC92,BCC95,CFS}.
