If you swing an object such as a bucket on a string, it undergoes both circular and rotational motion, and you should take both into account when performing energy calculations.
By circular motion, I mean motion about the centre of the circle. By rotational motion, I mean motion around the bucket's centre of mass.
If the energy due to the rotational motion of the object is very small compared with the energy due to the circular motion, then you can ignore it. That would be the case if the object is point like, or if the tangential velocity is very large compared with the angular velocity (ie if the string is very long).
If you are struggling to understand the difference, consider the following example. Imagine two identical discs of mass 1kg each glued on a circular table, 50cm from the centre of the table. If the table is made to spin about its centre, the discs will each move in the same way and will each have the same kinetic energy. Now, imagine that only one disc is glued to the table, and that the other sits on a frictionless axle sticking up from the table top. The movement of the two discs will no longer be the same. Both discs will follow a circular path around the centre of the table, but the glued disc will also rotate about its centre of mass, while the disc on the axle will not. As a consequence, the energy of the glued disc will be higher than the disc that is free to maintain its original orientation.