A slightly messy mesh in blender 2.81 beta I'm following the famous tutorial of the donut and the coffee cup from Blender Guru. I just noticed that some vertices of my object are slightly out of place. 
I don't know how that happened but I guess it happened after creating the edge loops. Could someone tell me how to correct the position of the vertices exactly so that they are aligned with the rest of the others?

On the other hand I have noticed that in a part of the cup there is a strange flattening. Anyone know why that is?
In second case, try: Edit mode, vertex, A, alt+m -> Merge by distace
Merge should work. Take a closer look for double vertex.
In first case:
Option one, If the vertex was displaced only by z coordinate, I would just simply copy/pasted it from correct vertex. Or use this tutorial: https://www.youtube.com/watch?v=TnTFl51GGdw
Final thought: you can just select the mirror vertex and copy it's coordinates to the vertex you need to correct.
So I would just select the vertex in the correct spot on other side of the handle/x-coordinate, press N to bring out the side panel, go to item, copy coordinates and paste them into coordinates of corrupted vertex one by one, BUT changing the x coordinate from positive to negative/from negative to positive.