An intense radiation field can modify plasma properties and the corresponding refractive index and lead to nonlinear propagation effects such as self-focusing. We estimate the corresponding effects in pair plasmas for circularly polarized waves, in both unmagnetized and strongly magnetically dominated cases. First, in the unmagnetized pair plasma the ponderomotive force does not lead to charge separation but to density depletion. Second, for astrophysically relevant plasmas of pulsar magnetospheres [and possible loci of fast radio bursts (FRBs)], where the cyclotron frequency ω_{B} dominates over the plasma frequency ω_{p} and the frequency of the electromagnetic wave ω_{B}≫ω_{p},ω, we show that (i) there is virtually no nonlinearity due to changing effective mass in the field of the wave; (ii) the ponderomotive force is F_{p}^{(B)}=-m_{e}c^{2}/4B_{0}^{2}∇E^{2}, which is reduced by a factor (ω/ω_{B})^{2} if compared to the unmagnetized case (B_{0} is the external magnetic field and E is the electric field of the wave); and (iii) for a radiation beam propagating along a constant magnetic field in the pair plasma with density n_{±}, the ponderomotive force leads to the appearance of circular currents that lead to a decrease of the field within the beam by a factor ΔB/B_{0}=2πn_{±}m_{e}c^{2}E^{2}/B_{0}^{4}. Applications to the physics of FRBs are discussed; we conclude that for the parameters of FRBs, the dominant magnetic field completely suppresses nonlinear self-focusing or filamentation.