Discussion of the equatorial photon motion in Kerr-Newman black-hole and naked-singularity spacetimes with a non-zero cosmological constant is presented. Both repulsive and attractive cosmological constants are considered. An appropriate `effective potential' governing the photon radial motion is defined, circular photon orbits are determined, and their stability with respect to radial perturbations is established. The spacetimes are divided into separated classes according to the properties of the `effective potential'. There is a special class of Kerr-Newman-de Sitter black-hole spacetimes with the restricted repulsive barrier. In such spacetimes, photons with high positive and all negative values of their impact parameter can travel freely between the outer black-hole horizon and the cosmological horizon due to an interplay between the rotation of the source and the cosmological repulsion. It is shown that this type of behaviour of the photon motion is connected to an unusual relation between the values of the impact parameters of the photons and their directional angles relative to the outward radial direction as measured in the locally non-rotating frames. Surprisingly, some photons counterrotating in these frames have a positive impact parameter. Such photons can be both escaping or captured in the black-hole spacetimes with the restricted repulsive barrier. For the black-hole spacetimes with a standard, divergent repulsive barrier of the equatorial photon motion, the counterrotating photons with positive impact parameters must all be captured from the region near the black-hole outer horizon as in the case of Kerr black holes, while they all escape from the region near the cosmological horizon. Further, the azimuthal motion is discussed and photon trajectories are given in typical situations. It is shown that for some photons with negative impact parameter turning points of their azimuthal motion can exist.