We investigate the potential of ground-based gravitational-wave detectors to probe the mass function of intermediate-mass black holes (IMBHs) wherein we also include BHs in the upper mass gap $sim 60-130~M_odot$. Using the noise spectral density 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 $50mbox{--} 500 M_odot$, 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 $2sigma$, the source-frame mass of the heavier component of the 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, we first evolve stars with massive helium cores using the open-source MESA software instrument to establish the upper and lower edges of the mass gap. We determine that the lower edge of the mass gap is $simeq$ 59$^{+34}_{-13}$ $M_{odot}$, while the upper edge is $simeq$ 139$^{+30}_{-14}$ $M_{odot}$, where the error bars indicate the mass range that follows from the $pm 3sigma$ uncertainty in the ${}^{12}text{C}(alpha, gamma) {}^{16} text{O}$ nuclear rate. 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. In this context, we also re-analyze the GW190521 event and show that the 90$%$ confidence interval of the primary-mass measurement lies inside the mass gap. Finally, we show that the precision achieved with the O4 run (and future O5 run) could be crucial for understanding the mass function, the formation mechanism, and evolution history of IMBHs.
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