Would the hoarseness from a vocal chord tumour be constant and consistent? or would it be varying?

Both: Probably varying early, then constant. See ENT surgeon for exam if you are consistently hoarse. He/she can look at vocal cords with scope and determine cause.

Consistant: The voice box is divided into two components, the extrinsic and intrinsic larynx. The latter refers to lesions directly on vocal chords, the former to those growths on the box which can invade chords depending where the primary is located. Once the chords are involved with a lesion they slowly lose vibrating capability which progresses until the lesion extends into the box itself .