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I am working on a flywheel currently, and I was wondering what is the force needed to recenter the flywheel when it is spinning and it is nudged slightly off axis? Is it related to gyroscopic precession?

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For this answer I'll assume "off-center" means the axis of rotation differs from the desired axis of rotation by angle $\theta$

The goal is to realign the axis of rotation, which can be accomplished by changing the rotational momentum vector.

This goal can be achieved by applying forces to the axle.

Fly Wheel force diagram

Applying symmetric forces as shown of magnitude $F$ a distance of $R$ from the center will add rotational inertia at a rate of $2FR$ Which will rotate the axis of the Gyroscope at angular rotation rate of $\frac{2FR}{I\omega}$ the axis of which will be perpendicular to the axis of the flywheel and parallel to the force being applied.

This process will not change the magnitude of the angular velocity of the flywheel.

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