A new subevent cumulant method was recently developed, which can significantly reduce the non-flow contributions in long-range correlations for small systems compared to the standard cumulant method. In this work, we study multi-particle cumulants in $p$+Pb collisions at $\sqrt{s_{\mathrm{NN}}} = 5.02$ TeV with a multiphase transport model (AMPT), including two- and four-particle cumulants ($c_{2}\{2\}$ and $c_{2}\{4\}$) and symmetric cumulants [SC(2, 3) and SC(2, 4)]... Our numerical results show that $v_{2}\{2\}$ is consistent with the experimental data, while the magnitude of $c_{2}\{4\}$ is smaller than the experimental data, which may indicate either the collectivity is underestimated or some dynamical fluctuations are absent in the AMPT model. For the symmetric cumulants, we find that the results from the standard cumulant method are consistent with the experimental data, but those from the subevent cumulant method show different behaviors. The results indicate that the measurements from the standard cumulant method are contaminated by non-flow effects, especially when the number of produced particles is small. The subevent cumulant method is a better tool to explore the $real$ collectivity in small systems. (read more)