We study e+e- --> pi+pi-h_c at center-of-mass energies from 3.90 GeV to 4.42 GeV using data samples collected with the BESIII detector operating at the Beijing Electron Positron Collider. The Born cross sections are measured at 13 energies, and are found to be of the same order of magnitude as those of e+e- --> pi+pi-J/psi but with a different line shape. In the pi^pm h_c mass spectrum, a distinct structure, referred to as Z_c(4020), is observed at 4.02 GeV/c^2. The Z_c(4020) carries an electric charge and couples to charmonium. A fit to the pi^pm h_c invariant mass spectrum, neglecting possible interferences, results in a mass of (4022.9pm 0.8pm 2.7) MeV/c^2 and a width of (7.9pm 2.7pm 2.6) MeV for the Z_c(4020), where the first errors are statistical and the second systematic. No significant Z_c(3900) signal is observed, and upper limits on the Z_c(3900) production cross sections in pi^pm h_c at center-of-mass energies of 4.23 and 4.26 GeV are set.
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