Light is composed of photons so we could ask if the photon has mass. Even before it was known that light is composed of photons it was known that light carries momentum and will exert a pressure on a surface. This is not evidence that it has mass since momentum can exist without mass.
Relativistic mass is a measure of the energy E of a particle which changes with velocity.
By convention relativistic mass is not usually called the mass of a particle in contemporary physics so it is wrong to say the photon has mass in this way.