I wonder should I precede the noun with the article 'a', but I don't think that is necessary?
I put [a] new carpet on the floor.
Reason: Carpet is a mass noun.
Am I correct?
It depends on what kind of carpet you mean.
If you put “a carpet” on the floor, I would interpret that to mean “a rug”: A thing that's laid out on the floor, on top of another floor covering, without being permanently attached to it.
If, however, the carpet is permanently attached to the floor, especially if it fills its room “wall-to-wall”, then that's the mass noun “carpet” (no article).