  At $x\approx0.25$
$\mathrm{Ba}_{1-x}\mathrm{K}_{x}\mathrm{Fe}_{2}\mathrm{As}_{2}$ undergoes a
novel first-order transition from a four-fold symmetric double-Q magnetic phase
to a two-fold symmetric single-Q phase, which was argued to occur
simultaneously with the onset of superconductivity (B\"ohmer et al., Nat. Comm.
6, 7911 (2015)). Here, by applying magnetic fields up to 10T, we investigate in
more detail the interplay of superconductivity with this magneto-structural
transition using a combination of high-resolution thermal-expansion and
heat-capacity measurements. We find that a magnetic field suppresses the
reentrance of the single-Q orthorhombic phase more strongly than the
superconducting transition, resulting in a splitting of the zero-field
first-order transition. The suppression rate of the orthorhombic reentrance
transition is stronger for out-of-plane than for in-plane fields and scales
with the anisotropy of the superconducting state. These effects are captured
within a phenomenological Ginzburg-Landau model, strongly suggesting that the
suppression of the reentrant orthorhombic single-Q phase is primarily linked to
the field-induced weakening of the superconducting order. Not captured by this
model is however a strong reduction of the orthorhombic distortion for
out-of-plane fields, which deserves further theoretical attention.
