I would only want cloth attached to the palm of hand and the rest of cloth not attached. I'm a little confused regarding weight painting and creating new vertex group for the cloth to attach to hand but I am a little overwhelmed with the process workflow, constraints etc. I sort of got it working, however cloth isn’t dragging properly and it seems to crumple up around hand.
With help of the Cloth settings > Shape > Pin group, you can pin one or more vertices of the cloth. Looks like you already did that. The vertices in the group are not affected by the cloth simulation but can be controlled by an animation. This way you can grab the cloth and drag it around.
In the example below the vertex in the center is assigned to a vertex group (Pin group) and this group is used as the pin group in the cloth settings.
The cloth is a default plane, subdivided 2 times. Additionally, it has a level 3 Subdivision Surface modifier to create more vertices for the cloth simulation. Then the Cloth modifier follows on the stack and another level 1 Subdivision Surface modifier to smooth the result.
To control the cleaning rag an armature with a single bone is used. The cleaning rag is bound to the bone of its armature (Ctrl+P Parent To > Bone). Animated bones have the advantage that you can create an action and use the action for different objects and use it in the NLA editor.
A Child Of Bone Constraint is used to attach the cleaning rag to the model's hand. (It can be added in Pose mode to the bone, then press the Set Inverse button). With help of the Influence value and keyframed Location you can detach the rag during the animation to put it away.
The arm model and the floor have a Collision modifier. If your cloth crumbles or crawls around the hand or floor you need to adjust the Collision distance settings in the Cloth settings and in the Collision settings. The settings depend on your model's size. Make sure you have applied the scale to the objects and there are no flipped normals.