  For a quantum system undergoing a unitary process work is commonly defined
based on the Two Projective Measurement (TPM) protocol which measures the
energies of the system before and after the process. However, it is well known
that projective measurements disregard quantum coherences of the system with
respect to the energy basis, thus removing potential quantum signatures in the
work distribution. Here we consider weak measurements of the system's energy
difference and establish corrections to work averages arising from initial
system coherences. We discuss two weak measurement protocols that couple the
system to a detector, prepared and measured either in the momentum or the
position eigenstates. Work averages are derived for when the system starts in
the proper thermal state versus when the initial system state is a pure state
with thermal diagonal elements and coherences characterised by a set of phases.
We show that by controlling only the phase differences between the energy
eigenstate contributions in the system's initial pure state, the average work
done during the same unitary process can be controlled. By changing the phases
alone one can toggle from regimes where the systems absorbs energy, i.e. a work
cost, to the ones where it emits energy, i.e. work can be drawn. This suggests
that the coherences are additional resources that can be used to manipulate or
store energy in a quantum system.
