You have that wrong in the momentum of massless particle has mass, independently of its carrier. Massless particles do not actually exist, what exist are particles with zero "mass at rest" and that are never at rest.
Yes, using it as the relativistic mass leads to confusing papers and equations, so it's better avoided. Yet, the press does that, what makes everything very confusing for lay people. I should have clarified that the OP usage of the term wasn't optimal (although common on the media he sees), but this following discussion about word definitions seems to be completely missing the point.
https://en.wikipedia.org/wiki/Energy%E2%80%93momentum_relati...
Besides it being a basic relation for relativity, there's also an entire "parallel" quantum mechanics for particles that have too much energy at the momentum component based on it.