  Using ground-based gravitational-wave detectors, we probe the mass function
of intermediate-mass black holes (IMBHs) wherein we also include BHs in the
upper mass gap $\sim 60-130~M_\odot$. Employing the projected sensitivity of
the upcoming LIGO and Virgo fourth observing (O4) run, we perform Bayesian
analysis on quasi-circular non-precessing, spinning IMBH binaries (IMBHBs) with
total masses $50\mbox{--} 500\, M_\odot$, mass ratios 1.25, 4, and 10, and
dimensionless spins up to 0.95, and estimate the precision with which the
source-frame parameters can be measured. We find that, at $2\sigma$, the mass
of the heavier component of IMBHBs can be constrained with an uncertainty of
$\sim 10-40\%$ at a signal-to-noise ratio of $20$. Focusing on the stellar-mass
gap with new tabulations of the $^{12}\text{C}(\alpha, \gamma)^{16} \text{O}$
reaction rate and its uncertanties, we evolve massive helium core stars using
\MESA\, to establish the lower and upper edge of the mass gap as
$\simeq$\,59$^{+34}_{-13}$\,$M_{\odot}$ and
$\simeq$\,139$^{+30}_{-14}$\,$M_{\odot}$ respectively, where the error bars
give the mass range that follows from the $\pm 3\sigma$ uncertainty in the
$^{12}\text{C}(\alpha, \gamma) ^{16} \text{O}$ nuclear reaction rate. We find
that high resolution of the tabulated reaction rate and fine temporal
resolution are necessary to resolve the peak of the BH mass spectrum. We then
study IMBHBs with components lying in the mass gap and show that the O4 run
will be able to robustly identify most such systems. Finally, we re-analyse
GW190521 with a state-of-the-art aligned-spin waveform model, finding that the
primary mass lies in the mass gap with 90\% credibility.
