We search for the process e+e-→χcJπ+π- (J=0, 1, 2) and for a charged charmoniumlike state in the χcJπ± subsystem. The search uses datasets 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 χcJπ+π- signals are observed at any center-of-mass energy, and thus upper limits are provided which also serve as limits for a possible charmoniumlike structure in the invariant χcJπ± mass. © 2021 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the "https://creativecommons.org/licenses/by/4.0/"Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by SCOAP3.