Impact of the in-medium cross section on cluster spectra in ${}^{40,48}\mathrm{Ca}+{}^{58,64}\mathrm{Ni}$ collisions at $56$ and $140$ $\mathbf{\mathrm{MeV}}/\mathrm{\mathbf{nucleon}}$

Although significant efforts have been made to investigate the density dependence of the nuclear symmetry energy, the influence of the in-medium cross section on particle production in transport models is not well constrained. The in-medium cross section reflects the dynamic situation of the medium such as a nontrivial phase space distribution. In this study, we analyze the transverse momentum spectra of $p$, $d$, $t$, ${}^3{\mathrm{He}}$ and $α$ particles emitted near mid-rapidity in central $^{40,48}\mathrm{Ca}$ + $^{58, 64}\mathrm{Ni}$ reactions at $56$ and $140$ $\mathrm{MeV}/\mathrm{nucleon}$. The Antisymmetrized Molecular Dynamics ($\mathrm{AMD}$) model is chosen as the transport model for data comparison. Central events are selected based on charged-particle multiplicity in both the experimental data and AMD calculations after applying an experimental filter. Our results show that the in-medium nucleon-nucleon scattering cross-sections are more strongly reduced at $56$ $\mathrm{MeV}/\mathrm{nucleon}$ than at $140$ $\mathrm{MeV}/\mathrm{nucleon}$ incident energy.