  Spin-induced quadrupole moments provide an important characterization of
compact objects, such as black holes, neutron stars and black hole mimickers
inspired by additional fields and/or modified theories of gravity. Black holes
in general relativity have a specific spin-induced quadrupole moment, with
other objects potentially having differing values. Different values of this
quadrupole moment lead to modifications of the spin precession dynamics, and
consequently modifications to the inspiral waveform. Based on the spin-dynamics
and the associated precessing waveform developed in our previous work, we
assess the prospects of measuring spin-induced moments in various black hole,
neutron star, and black-hole mimicker binaries. We focus on binaries in which
at least one of the objects is in the mass gap (similar to the $2.6 M_\odot$
object found in GW190814). We find that for generic precessing binaries, the
effect of the spin-induced quadrupole moments on the precession is sensitive to
the nature of the mass-gap object, i.e., whether it is a light black hole or a
massive neutron star. So that this is a good probe of the nature of these
objects. For precessing black-hole mimicker binaries, this waveform also
provides significantly tighter constraints on their spin-induced quadrupole
moments than the previous results obtained without incorporating the precession
effects of spin-induced quadrupole moments. We apply the waveform to sample
events in GWTC catalogs to obtain better constraints on the spin-induced
quadrupole moments, and discuss the measurement prospects for events in the
O$4$ run of the LIGO-Virgo-KAGRA Collaboration.
