Cross sections for the reactions $e^+e^-\rightarrow K^+K^-π^+π^-(π^0)$, $K^+K^-K^+K^-(π^0)$, $π^+π^-π^+π^-(π^0)$, $p\bar{p}π^+π^-(π^0)$ in the energy region between 3.773 and 4.600 GeV

Abstract: Using the data samples collected in the energy range from 3.773 to 4.600 GeV with the BESIII detector at the BEPCII collider, we measure the dressed cross sections as a function of center-of-mass energy for $$e^{+e^-\rightarrow} K<sup>+K<sup>-\pi<sup>+\pi<sup>-(\pi<sup>0)$$, $K<sup>+K<sup>-K<sup>+K<sup>-(\pi<sup>0)$, $\pi<sup>+\pi<sup>-\pi<sup>+\pi<sup>-(\pi<sup>0)$, and $p\bar{p}\pi<sup>+\pi<sup>-(\pi<sup>0)$. The cross sections for $$e^{+e^-\rightarrow} K<sup>+K<sup>-K<sup>+K<sup>-\pi<sup>0$$, $p\bar{p}\pi<sup>+\pi<sup>-(\pi<sup>0)$ are the first measurements. Cross sections for the other five channels are much more precise than previous results in this energy region. We also search for charmonium and charmonium-like resonances, such as the $Y(4230)$, decaying into the same final states. We find evidence of the $\psi(4040)$ decaying to $\pi<sup>+\pi<sup>-\pi<sup>+\pi<sup>-\pi<sup>0$ with a statistical significance of $3.6\sigma$. Upper limits are provided for other decays since no clear signals are observed.