  We investigate the stochastic dynamics of a thermal machine realized by a
fast-driven Otto cycle. By employing a stochastic approach, we find that system
coherences strongly affect fluctuations depending on the thermodynamic current.
Specifically, we observe an increment in the system instabilities when
considering the heat exchanged with the cold bath. On the contrary, the cycle
precision improves when the system couples with the hot bath, where
thermodynamic fluctuations reduce below the classical Thermodynamic Uncertainty
Relation bound. Violation of the classical bound holds even when a dephasing
source couples with the system. We also find that coherence suppression not
only restores the cycle cooling but also enhances the convergence of
fluctuation relations by increasing the entropy production of the reversed
process. An additional analysis unveiled that the stochastic sampling required
to ensure good statistics increases for the cooling cycle while downsizes for
the other protocols. Despite the simplicity of our model, our results provide
further insight into thermodynamic relations at the stochastic level.
