  Extreme-mass-ratio inspirals (EMRIs) will be key sources for LISA. However,
accurately extracting system parameters from a detected EMRI waveform will
require self-force calculations at second order in perturbation theory, which
are still in a nascent stage. One major obstacle in these calculations is the
strong divergences that are encountered on the worldline of the small object.
Previously, it was shown by one of us [Phys. Rev. D 95, 104056 (2017)] that a
class of "highly regular" gauges exist in which the singularities have a
qualitatively milder form, promising to enable more efficient numerical
calculations. Here we derive expressions for the metric perturbation in this
class of gauges, in a local expansion in powers of distance $r$ from the
worldline, to sufficient order in $r$ for numerical implementation in a
puncture scheme. Additionally, we use the highly regular class to rigorously
derive a distributional source for the second-order field and a pointlike
second-order stress-energy tensor (the Detweiler stress-energy) for the small
object. This makes it possible to calculate the second-order self-force using
mode-sum regularisation rather than the more cumbersome puncture schemes that
have been necessary previously. Although motivated by EMRIs, our calculations
are valid in an arbitrary vacuum background, and they may help clarify the
interpretation of point masses and skeleton sources in general relativity more
broadly.
