As you see in the description of the Horklump, it is a mobile creature.
The Horklump can cover an average garden in a matter of days.
It is a common mistake that people think it grows to cover the garden, but according to Pottermore, it infests the garden, it does not grow to cover the garden.
Therefore it is a mobile creature.
The definition of plant is,
A living organism of the kind exemplified by trees, shrubs, herbs, grasses, ferns, and mosses, typically growing in a permanent site, absorbing water and inorganic substances through its roots, and synthesizing nutrients in its leaves by photosynthesis using the green pigment chlorophyll.
Although this is a "Muggle" definition, it carries over to the wizarding world. The Venomous Tentacula is stationary, and so is the Mandrake. Although they do move, their roots stay in the same place. Therefore, the Horklump is then classified as a beast by it's ability to move to different locations, where other wizarding plants are stationary. Also, as this beast searches for food, it leads us to believe that it does not go through photosynthesis, and that it's main source of food are the worms that it finds.
It spreads it's sinewy tentacles rather than roots into the ground to search for it's favorite food of earthworms.