The thermopower, S, of La_0.7Ca_0.3Mn_1-yFe_yO₃ (y = 0-0.09), measured for T = 25-310 K and magnetic fields of B = 0-10 T, exhibits strong sensitivity to doping with Fe. The weak increase of S observed in the undoped material with lowering temperature is enhanced considerably in the sample with y = 0.03 until a steep decrease takes place in the vicinity of the paramagnetic-ferromagnetic transition temperature, T_C. S increases with further growth of y and its maximum follows the decrease of T_C from 228 K at y = 0.03 to 119 K at y = 0.09. Application of a magnetic field reduces S strongly. The experimental data are analysed quantitatively with a percolation model using results of our recent investigations of the Shklovskii-Efros-like variable-range hopping conductivity in the same set of La_1-xCa_xMn_1-yFe_yO₃ samples with evidence for a soft Coulomb gap and a rigid gap in the density of states near the Fermi level (Laiho et al 2002 J. Phys.: Condens. Matter 14 8043). In the paramagnetic insulating phase the temperature dependence of S is determined by competing contributions from two types of asymmetry of the gap: a small shift of its centre with respect to the Fermi level and cubic non-parabolicity. Numerical values of the shift and the cubic asymmetry terms are determined.