  We present a calculation of $H+j$ at NLO including the effect of a finite
top-mass. Where possible we include the complete dependence on $m_t$. This
includes the leading order amplitude, the infrared poles of the two-loop
amplitude and the real radiation amplitude. The remaining finite piece of the
virtual correction is considered in an asymptotic expansion in $m_t$, which is
accurate to $m_t^{-4}$. By successively including more $m_t$-exact pieces, the
dependence on the asymptotic series diminishes and we find convergent behavior
for $p_T^H>m_t$ for the first time. Our results justify rescaling by the
$m_t$-exact LO cross section to model top-mass effects in EFT results up to
$p_T$ of 250 to 300 GeV. We show that the error made by using the LO rescaling
becomes comparable to the NNLO scale uncertainty for such large energies. We
implement our results into the Monte Carlo code MCFM.
