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Let's assume an exoplanet orbits a red dwarf star closely enough that is tidally locked to it. Is it possible for the planet to have a significant axial precession? If not naturally, is it possible to strike the planet with a glancing blow of something akin to a rogue planet that this precession is imparted?

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Yes.

Our own Moon is tidally locked to the Earth, but has an axial precession with a period of about 18.6 years. This precession is due to the 1.5 degree offset angle between the Moon's rotational axis and the ecliptic (the Earth's orbital plane around the Sun).

What does that have to do with your question? If instead of a Sun/Earth/Moon orbital system, suppose there was a similar black-hole/red-dwarf/exoplanet system, in which the exoplanet's rotational axis was offset from the red dwarf's orbital plane around the black hole. Then the exoplanet would exhibit axial precession despite being tidally locked to the red dwarf. In a simulation, one could increase the axial precession effects by increasing the masses of the bodies, decreasing the distances between the bodies, or increasing the angle between the exoplanet's rotational axis and the red dwarf's orbital plane.

I don't think axial precession could be directly induced by a glancing blow from a rogue planet. This is because axial precession requires a consistent mechanism over time. However, if the blow imparted rotational momentum that increased the offset angle of axis to ecliptic, the blow could certainly indirectly cause or increase axial precession.

Note: this answer doesn't rule out other potential mechanisms for the OP's scenario, such as extreme oblateness of the red dwarf.

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