The relativistic motion of a charged particle in a superposition of circularly polarized plane electromagnetic waves and a uniform magnetic field is studied by deriving an exact solution to the Lorentz force equation of the charged particle. All of the circularly polarized plane electromagnetic waves propagate parallel to the uniform magnetic field. The explicit expressions of the charged particle position, velocity, and energy are obtained for arbitrary initial conditions, and the behavior of the particle motion is studied both analytically and numerically. It is found that for certain initial conditions, the particle gains energy from the waves and the energy gain could reach a maximum value during the time evolution of the charged particle motion in the nonresonance case. In addition, the particle could be accelerated to a much higher energy at the cyclotron resonance, and, in comparison with the situation of a single wave, the use of the superposition wave that consists of a group of circularly polarized plane electromagnetic waves with different frequencies increases the chance of the occurrence of cyclotron resonance for a charged particle, making it easier for the particle to be accelerated. It has also been observed that the interaction of the charged particle with the superposition of electromagnetic waves can be improved significantly at cyclotron resonance when the frequency and phase differences of the waves remain so small that the phases of the waves are in the same quadrant. The results of the present paper are of interest to particle acceleration and heating applications, as well as to basic plasma processes.