  We present a generator for the production of a Higgs boson H in association
with a vector boson V=W or Z (including subsequent V decay) plus zero and one
jet, that can be used in conjunction with general-purpose shower Monte Carlo
generators, according to the POWHEG method, as implemented within the POWHEG
BOX framework. We have computed the virtual corrections using GoSam, a program
for the automatic construction of virtual amplitudes. In order to do so, we
have built a general interface of the POWHEG BOX to the GoSam package. With
this addition, the construction of a POWHEG generator within the POWHEG BOX is
now fully automatized, except for the construction of the Born phase space. Our
HV + 1 jet generators can be run with the recently proposed MiNLO method for
the choice of scales and the inclusion of Sudakov form factors. Since the HV
production is very similar to V production, we were able to apply an improved
MiNLO procedure, that was recently used in H and V production, also in the
present case. This procedure is such that the resulting generator achieves NLO
accuracy not only for inclusive distributions in HV + 1 jet production but also
in HV production, i.e. when the associated jet is not resolved, yielding a
further example of matched calculation with no matching scale.
