The self-force is the leading method in modelling waveforms for extreme mass
ratio inspirals, a key target of ESA's future space-based gravitational wave
detector LISA. In modelling these systems, one approximates the smaller body as
a point particle leading to problematic singularities that need to be removed.
Modelling of this singular structure has settled on the Detweiler-Whiting
singular field as the gold standard. As a solution to the governing wave
equation itself, on removal, it leaves a smooth regular field that is a
solution to the homogeneous wave equation, much like its well established flat
spacetime counterpart. The mode-sum method enables subtraction of this
singularity mode by mode via a spherical harmonic decomposition. The more modes
one has, the faster the convergence in the $\ell $ -sum, making these expressions
highly beneficial, especially considering the heavy computational burden of
waveform production. Until recently, only the two leading orders were known for
generic orbits in Kerr spacetime. In a previous paper, we produced the next
non-zero parameter for a scalar charged particle in curved spacetime, laying
the groundwork for the electromagnetic and gravitational case which we present
here.