  We consider the production of a vector boson ($Z$, $W^\pm$ or $\gamma^*$) at
next-to-next-to-leading order in the strong coupling constant $\alpha_{\rm S}$.
  We impose a transverse-momentum cutoff, $q_{\rm T}^{\rm cut}$, on the vector
boson produced in the $qg$-initiated channel. We then compute the power
corrections in the cutoff, up to the second power, of the real-virtual
interference contribution to the cumulative cross section at order $\alpha_{\rm
S}^2$. Other terms with the same kinematics, originating from the subtraction
method applied to the double-real contribution, have been also considered.
  The knowledge of such power corrections is a required ingredient in order to
reduce the dependence on the transverse-momentum cutoff of the QCD cross
sections at next-to-next-to-leading order, when the $q_{\rm T}$-subtraction
method is applied. In addition, the study of the dependence of the cross
section on $q_{\rm T}^{\rm cut}$ allows as well for an understanding of its
behaviour in the small transverse-momentum limit, giving hints on the structure
at all orders in $\alpha_{\rm S}$ and on the identification of universal
patterns.
  Our result are presented in an analytic form, using the process-independent
procedure described in a previous paper for the calculation of the all-order
power corrections in $q_{\rm T}^{\rm cut}$.
