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We present 20 years of timing observations for 1E 1207.4-5209, the central compact object in supernova remnant PKS 1209-51/52, to follow up on our detection of an unexpected timing glitch in its spin-down. Using new XMM-Newton and NICER observations of 1E 1207.4-5209, we now find that the phase ephemeris can be well modelled by either two small glitches, or extreme timing noise. The implied magnitudes of the frequency glitches are Delta f/f = (9+\-2)E-10 and Delta f/f = (3.7+/-0.7)E-10, at epochs 2010.9 and 2014.4, respectively. The updated timing solutions also rule out our previous suggestion of a large glitch in the frequency derivative fdot. No other canonical pulsar with such a small spin-down rate (fdot = -1.2E-16 Hz/s) or surface dipole magnetic field strength (B_s = 9.8E10 G) has been observed to glitch; the glitch activity parameter of 1E 1207.4-5209 is larger than that of more energetic pulsars. Alternative parameterizations that do not involve glitches can fit the data, but they have timing residuals or a second frequency derivative fddot that are orders of magnitude larger than in pulsars with similar spin-down parameters. These timing properties of 1E 1207.4-5209 further motivate the leading theory of central compact objects, that an initial B-field of normal strength was buried in the neutron star crust by fallback of supernova ejecta, suppressing the surface dipole field. The slow reemergence of the buried field may be involved in triggering glitches or excess timing noise.