When moving in the same plane as gravity, that is, perpendicular to the ground, gravity MINUS the force on the rope is centripetal force. Gravity and rope force are not in the same direction, but vectors.

When a rope pulls an object around the crown block on a smooth horizontal surface, the pulling force on the rope is the centripetal force.

When the rope pulls an unsmooth object around the crown block in a uniform circular motion on an unsmooth horizontal plane, the gravity on the object is the pressure f on the ground, multiplied by the sliding friction coefficient μ between the object and the ground, which is the friction force. But in order to make an object move in a uniform circle, there must be a force to counteract the friction, which will not affect the magnitude and direction of the tension on the rope.

Like we usually pull a rope with our hands to help things turn, in fact, it is not a circular motion.