  With recent advances in asteroseismology it is now possible to peer into the
cores of red giants, potentially providing a way to study processes such as
nuclear burning and mixing through their imprint as sharp structural variations
-- glitches -- in the stellar cores. Here we show how such core glitches can
affect the oscillations we observe in red giants. We derive an analytical
expression describing the expected frequency pattern in the presence of a
glitch. This formulation also accounts for the coupling between acoustic and
gravity waves. From an extensive set of canonical stellar models we find
glitch-induced variation in the period spacing and inertia of non-radial modes
during several phases of red-giant evolution. Significant changes are seen in
the appearance of mode amplitude and frequency patterns in asteroseismic
diagrams such as the power spectrum and the \'echelle diagram. Interestingly,
along the red-giant branch glitch-induced variation occurs only at the
luminosity bump, potentially providing a direct seismic indicator of stars in
that particular evolution stage. Similarly, we find the variation at only
certain post-helium-ignition evolution stages, namely, in the early phases of
helium-core burning and at the beginning of helium-shell burning signifying the
asymptotic-giant-branch bump. Based on our results, we note that assuming stars
to be glitch-free, while they are not, can result in an incorrect estimate of
the period spacing. We further note that including diffusion and mixing beyond
classical Schwarzschild, could affect the characteristics of the glitches,
potentially providing a way to study these physical processes.
