  Recently, upper bounds on the static time-reversal ($\mathcal{T}$) and
spatial parity ($\mathcal{P}$)-violating electron electric dipole moment
($e$EDM) and dimensionless constant characterizing the strength of the
$\mathcal{T}$,$\mathcal{P}$-violating scalar-pseudoscalar nucleus-electron
interaction have been updated in the JILA experiment using the HfF$^+$ cations
[T.S. Roussy et al, arxiv:2212.11841 (2022)]. We considered other two sources
of the $\mathcal{T}$,$\mathcal{P}$,-violation in HfF$^+$ --
axionlike-particle-mediated (ALP) scalar-pseudoscalar electron-electron and
nucleus-electron interactions. To estimate the magnitude of effects, induced by
such interactions in HfF$^+$ we have developed and applied a method which
implies direct use of the ab-initio relativistic coupled cluster theory to
calculate molecular parameters that characterize the interactions. Using these
parameters, we showed that an order of magnitude updated laboratory constraints
on the ALP-mediated electron-electron and nucleus-electron interactions can be
derived from the experimental data on $\mathcal{T}$,$\mathcal{P}$-violating
effects in HfF$^+$ for a wide range of ALP masses.
