  Resistance measurements on single crystals of CaK(Fe$_{1-x}$Mn$_x$)$_4$As$_4$
($x$ = 0.024) were performed under hydrostatic pressure up to 5.15 GPa. The
pressure dependence of the magnetic and superconducting transition temperatures
and that of the superconducting upper critical field are reported. Our results
show that upon increasing pressure, the magnetic transition temperature $T_{N}$
is suppressed, whereas the superconducting transition temperature $T_{c}$ first
increases and then decreases, exhibiting a maximum at a pressure $p_c$
corresponding to the intersection of the $T_{N}$($p$) and $T_{c}$($p$) lines.
In addition, a minimum in the normalized slope of the superconducting upper
critical field as well as a change in the pressure dependence of the inferred
superconducting coherence length are observed at $p_c$, suggesting a difference
in the Fermi surface of the paramagnetic and antiferromagnetic states. Finally,
CaK(Fe$_{1-x}$Mn$_x$)$_4$As$_4$ ($x$ = 0.024) likely goes through a
half-collapsed-tetragonal phase transition at $\sim$ 4.3 GPa, further
demonstrating that the half-collapsed-tetragonal transition pressure in the
CaKFe$_4$As$_4$ system is relatively insensitive to transition metal
substitution.
