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We report the transverse momentum $p_T$ spectra of identified hadrons ($π^\pm$, $K^\pm$ and $p(\bar p)$) in Au+Au collisions at $\sqrt{s_{NN}}$ = 7.7 - 39 GeV from A Multi Phase Transport Model with string melting effect (AMPT-SM). During this study, a new set of parameters are explored to study the effect of hadronic cascade by varying hadronic cascade time $t_{max}$ = 30 $f$m/$c$ and 0.4 $f$m/$c$. No significant effect of this change is observed in the $p_T$ spectra of light hadrons and the AMPT-SM model reasonably reproduces the experimental data. To investigate the kinetic freeze-out properties the blast wave fit is performed to the $p_T$ spectra and it is found that the blast wave model describes the AMPT-SM simulations well. We additionally observe that the kinetic freeze-out temperature ($T_{kin}$) increases from central to peripheral collisions, which is consistent with the argument of short-lived fireball in peripheral collisions. Whereas the transverse flow velocity, $<β_T>$ shows a decreasing trend from central to peripheral collisions indicating a more rapid expansion in the central collisions. Both, $T_{kin}$ and $<β_T>$ show a weak dependence on the collision energy at most energies. We also observe a strong anti-correlation between $T_{kin}$ and $<β_T>$. The extracted freeze-out parameters from the AMPT-SM simulations agree with the experimental data as opposed to earlier studies that reported some discrepancies. Whereas, no significant effect is found on the freeze-out parameters by varying the $t_{max}$. We also report the $p_T$ spectra of light hadrons and their freeze-out parameters by AMPT-SM simulations at $\sqrt{s_{NN}}$ = 14.5 GeV, where no experimental data is available for comparison. Overall, the set of parameters used in this study well describes the experimental data at BES energies.