  We search for the process $e^{+}e^{-}\rightarrow \pi ^{+}\pi ^{-} \chi_{cJ}$
($J=0,1,2$) and for a charged charmonium-like state in the $\pi ^{\pm}
\chi_{cJ}$ subsystem. The search uses data sets collected with the BESIII
detector at the BEPCII storage ring at center-of-mass energies between 4.18 GeV
and 4.60 GeV. No significant $\pi ^{+}\pi ^{-} \chi_{cJ}$ signals are observed
at any center-of-mass energy, and thus upper limits are provided which also
serve as limits for a possible charmonium-like structure in the invariant $\pi
^{\pm} \chi_{cJ}$ mass.
