  We study a schematic mode-coupling model in which the ideal glass transition
is cut off by a decay of the quadratic coupling constant in the memory
function. (Such a decay, on a time scale tau_I, has been suggested as the
likely consequence of activated processes.) If this decay is complete, so that
only a linear coupling remains at late times, then the alpha relaxation shows a
temporal crossover from a relaxation typical of the unmodified schematic model
to a final strongly slower-than-exponential relaxation. This crossover, which
differs somewhat in form from previous schematic models of the cut-off glass
transition, resembles light-scattering experiments on colloidal systems, and
can exhibit a `slower-than-alpha' relaxation feature hinted at there. We also
consider what happens when a similar but incomplete decay occurs, so that a
significant level of quadratic coupling remains for t>>tau_I. In this case the
correlator acquires a third, weaker relaxation mode at intermediate times. This
empirically resembles the beta process seen in many molecular glass formers. It
disappears when the initial as well as the final quadratic coupling lies on the
liquid side of the glass transition, but remains present even when the final
coupling is only just inside the liquid (so that the alpha relaxation time is
finite, but too long to measure). Our results are suggestive of how, in a
cut-off glass, the underlying `ideal' glass transition predicted by
mode-coupling theory can remain detectable through qualitative features in
dynamics.
