  We consider the renormalization group equations within the Standard Model
Effective Field Theory and compute two-loop contributions proportional to the
top quark Yukawa coupling for the operator generating an effective Higgs-gluon
coupling, focusing on the Yukawa-like operator. These two-loop running effects
are relevant for processes where the effective Higgs-gluon coupling contributes
at a lower loop order compared to the Standard Model contribution and where a
dynamical scale choice is adopted. Such a situation arises, for instance, in
the Higgs transverse momentum distribution and Higgs pair production. We
investigate the phenomenological impact of our computations on these two
processes and find that the two-loop contributions are significant and can lead
to deviations of up to 20\% in the scenarios we consider.
