Tip of my thumb nail is partially cut. should i remove it?

Remove: Partially cut nails should be clipped to prevent uneven growth, which could result in an ingrown mail.

Yes and no: If it is a small piece you can remove it. If it is covering part of the bed under the nail then keep it cover till it heals underneath. The bed under the nail is very tender if exposed. Apply vaseline to the area that is exposed till it heals.