  We consider a stochastic process undergoing resetting after which a random
refractory period is imposed. In this period the process is quiescent and
remains at the resetting position. Using a first-renewal approach, we compute
exactly the stationary position distribution and analyse the emergence of a
delta peak at the resetting position. In the case of a power-law distribution
for the refractory period we find slow relaxation. We generalise our results to
the case when the resetting period and the refractory period are correlated, by
computing the Laplace transform of the survival probability of the process and
the mean first passage time, i.e., the mean time to completion of a task. We
also compute exactly the joint distribution of the active and absorption time
to a fixed target.
