We introduce a perturbative method to calculate all moments of the
first-passage time distribution in stochastic one-dimensional processes which
are subject to both white and coloured noise. This class of non-Markovian
processes is at the centre of the study of thermal active matter, that is
self-propelled particles subject to diffusion. The perturbation theory about
the Markov process considers the effect of self-propulsion to be small compared
to that of thermal fluctuations. To illustrate our method, we apply it to the
case of active thermal particles (i) in a harmonic trap (ii) on a ring. For
both we calculate the first-order correction of the moment-generating function
of first-passage times, and thus to all its moments. Our analytical results are
compared to numerics.