  In this paper, we give quantitative bounds on the $f$-total variation
distance from convergence of an Harris recurrent Markov chain on an arbitrary
under drift and minorisation conditions implying ergodicity at a sub-geometric
rate. These bounds are then specialized to the stochastically monotone case,
covering the case where there is no minimal reachable element. The results are
illustrated on two examples from queueing theory and Markov Chain Monte Carlo.
