  We investigate the frame-dragging effect of the hairy Kerr spacetimes on the
spin of a test gyro and accretion disk. Firstly, we analyze Lense-Thirring (LT)
precession frequency, geodetic precession frequency, and the general spin
precession frequency of a test gyro attached to a stationary observer in the
spacetime. We find that the black hole hair suppresses those precession
frequencies in comparison with that occurs in Kerr spacetime in general
relativity. Moreover, using those frequencies as probe, we differentiate the
hairy Kerr black hole (BH) from naked singularity (NS). Specifically, as the
observer approaches the central source along any direction, the frequencies
grow sharply for the hairy Kerr BH, while for the hairy NS they are finite
except at the ring singularity. Then, we investigate the quasiperiodic
oscillations (QPOs) phenomena as the accretion disk approaches the hairy Kerr
BH or NS. To this end, we analyze the bound circular orbits and their
perturbations. We find that as the orbits approach the corresponding inner-most
stable circular orbit (ISCO), both LT precession frequency and periastron
precession frequency behave differently in the hairy Kerr BH and NS.
Additionally, the hairy parameters have significant effects on the two
frequencies. We expect that our theoretical studies could shed light on
astrophysical observations in distinguishing hairy theories from Einstein's
gravity, and also in distinguishing BH from NS in spacetime with hair.
