Circular movement

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In nature there doesn't exist perfect circular movement, yet you can always approximate what looks like a circular movement with a perfect one. My question is this:
When a circular movement is not perfect, say an object has a tangetial velocity of v-ε in a radius where a velocity of v is required for perfect circular movement and ε is a tiny number.
Will the object then move in an approximate circular movement? My intuition says of course because everyday we observe lots of approximate circular motions. However, something for me says no, because when the velocity is not perfectly right you can't apply the F$\bullet$v = 0 and that's mean that work will be done on the object towards the centre of rotation. Gradually then the object will move towards the centre. My question is - for this situation where the object has a velocity that is nearly sufficient for a circular motion - will the work done per time towards the centre of rotation be less than if the object was lying still and being pulled in by same inwards force.
Hope this made at least somewhat sense