  We develop the recently proposed analytical R-matrix (ARM) method to
encompass strong field ionisation by circularly polarised fields, for atoms
with arbitrary binding potentials. Through ARM, the effect of the potential can
now be included consistently both during and after ionisation, providing a
complete picture for the effects of the long-range potential. We find that the
Coulomb effects modify the ionisation dynamics in several ways, including
modification of (i) the ionisation (exit) times, (ii) the initial conditions
for the electron continuum dynamics, (iii) the "tunnelling angle", at which the
electron "enters" the barrier, and (iv) the electron drift momentum. We derive
analytical expressions for the Coulomb-corrected ionisation times, initial
velocities, momentum shifts and ionisation rates in circularly polarised
fields, for arbitrary angular momentum of the initial state. We also analyse
how the non-adiabatic Coulomb effects modify (i) the calibration of the
attoclock in the angular streaking method, and (ii) the ratio of ionisation
rates from $p^{-}$ and $p^{+}$ orbitals.
