  Clusters of galaxies possess the capability to accelerate cosmic rays (CRs)
to very high energy up to $\sim10^{18}$~eV due to their large size and magnetic
field strength which favor CR confinement for cosmological times. During their
confinement, they can produce neutrinos and $\gamma-$rays out of interactions
with the background gas and photon fields. In recent work,
\cite{hussain2021high, hussain2023diffuse} have conducted three-dimensional
cosmological magnetohydrodynamical (MHD) simulations of the turbulent
intracluster medium (ICM) combined with multi-dimensional Monte Carlo
simulations of CR propagation for redshifts ranging from $z \sim 5$ to $z = 0$
to study the multi-messenger emission from these sources. They found that when
CRs with a spectral index in the range $1.5 - 2.5$ and cutoff energy
$E_\mathrm{max} = 10^{16} - 10^{17}$~eV are injected into the system, they make
significant contributions to the diffuse background emission of both neutrinos
and gamma-rays. In this work, we have revisited this model and undertaken
further constraints on the parametric space. This was achieved by incorporating
the recently established upper limits on neutrino emission from galaxy
clusters, as obtained by the IceCube experiment. We find that for CRs injected
with spectral indices in the range $2.0 - 2.5$, cutoff energy $E_\mathrm{max} =
10^{16} - 10^{17}$~eV, and power corresponding to $(0.1-1)\%$ of the cluster
luminosity, our neutrino flux aligns with the upper limits estimated by
IceCube. Additionally, the resulting contribution from clusters to the diffuse
$\gamma$-ray background (DGRB) remains significant with values of the order of
$ \sim 10^{-5}\, \mathrm{MeV} \, \mathrm{cm}^{-2} \,\mathrm{s}^{-1} \,
\mathrm{sr}^{-1}$ at energies above $500$ GeV.
