Yes. A good understanding of what happens requires some careful thinking. Try this.
For an object moving in a vertical circle on a string, the 2 forces on the object are string's tension (T acting radially inwards along the string) and the object's weight (W acting vertically downwards).
During each rotation W remains constant but the magnitude and direction of T change.
Note, the directions of T and W are different except at the top position.
The resultant force (F) on the object is the *vector sum* of T and W.
The centripetal force (C) is the *radially inwards component* of F.
Therefore ‘gravity’ (weight, W) *is* a part of the centripetal force except at the points where the string is horizontal.
The magnitude of C only depends on mass, speed and radius. |C| = mv²/r. Speed is usually fastest at the bottom position but for simplicity imagine speed is constant, so |C| is constant.
At the top, T and W are in the same direction, both down so:
|C| = |T_top| + |W|.
At the bottom, T and W are in opposite directions, T upwards and W downwards so:
|C| = |T_bottom| - |W|
|T_top| + |W| = |T_bottom| - |W|
|T_bottom| - |T_top| = 2|W|
So the tension at the bottom is bigger than the tension at the top by and amount equal to twice the weight.
If the object is moving faster at the bottom than the top, this difference is even bigger.