  We demonstrate the effectiveness of averaging over the Wilson parameter r
(which has been proposed earlier) in removing the cutoff effects of naive
Wilson fermions in both the anomaly term and the pseudoscalar density term in
the flavor singlet axial Ward identity at O(g^2) involving slowly varying
background gauge fields. We show that it is the physical fermion contribution
which is largely influenced by the r averaging. We have studied the possible
interplay between finite size and cutoff effects by investigating in detail
naive, O(a) improved and OStm Wilson fermion cases for a range of volumes and
lattice fermion mass (am). For naive Wilson fermions r averaging is shown to
remove the effects of the interplay. We have shown that for the pseudoscalar
density term to O (g^2) the lattice result differs from the continuum result by
exhibiting considerable am dependence which appears to be a manifestation of
cutoff effects with naive Wilson fermion. The pseudoscalar density term to
O(g^2) is shown to be almost independent of am when r-averaging is performed.
