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We present precise predictions for the hadronic production of an on-shellHiggs boson in association with a leptonically decaying gauge boson and a jetup to order $\alpha_s^3$. We include the complete set of NNLO QCD correctionsto both charged- and neutral-current Drell-Yan type contributions, as well asthe previously known leading heavy quark loop induced contributions whichinvolve a direct Higgs-quark coupling. As an application, we study a range ofdifferential observables in proton-proton collisions at $\sqrt{s} =13~\text{TeV}$ for both the charged- and neutral-current production modes. Foreach Higgs production process, we assess the improvement in the theoreticaluncertainty for both the exclusive ($n_{\text{jet}} = 1$) and inclusive($n_{\text{jet}} \geq 1$) jet categories. We find that the inclusion of theNNLO corrections to the Drell-Yan type contributions is essential instabilising the predictions and in reducing the theoretical uncertainty forboth inclusive and exclusive jet production for all three modes. This isparticularly true in the kinematical regimes associated with low to mediumvalues of the transverse momentum of the produced vector boson and where thedifferential cross sections are the largest. For the neutral-current process,we find that the heavy quark loop induced contributions have their largestphenomenological impact (an increase in the size of the NNLO corrections, adistortion of the distribution shape and an enlargement of the left overremaining uncertainties) in kinematical regions associated to large values of$p_{T,Z}$ (typically above $150~\text{GeV}$) where the cross sections aresmaller.

$VH+\text{jet}$ production in hadron-hadron collisions up to order $\alpha_s^3$ in perturbative QCD