The combustion characteristics of three novel energetic cage hydrocarbon compounds - bis(nitratomethyl)-1,3bishomocubane (DNMBHC), diazido-dimethyl-bishomocubane (DADMBHC), and 1,3-bishomocubane dimer (BHCD), and an RP-1 surrogate fuel (RP-1-s) were investigated through droplet combustion experiments. As these novel compounds were envisioned as potential additives to RP-1 in semi-cryogenic engines, where liquid oxygen (LOX) would be used as the oxidizer, it was necessary to test these fuels in oxygen-enriched environments. Studies were conducted in two different environments - one containing 50% O2 and the other having 80% O2 with balance N2 in both cases. The influence of oxygen concentration on the variation of droplet and flame dimensions with time, time period of combustion, flame temperature, and sooting propensities were studied using videography and color-ratio pyrometry. With an increase in oxygen concentration, the flame temperatures were found to increase with a concurrent reduction in the flame stand-off ratios, resulting in faster combustion. As compared to trials in ambient air, the effect of microexplosions generated as a result of condensed phase pyrolysis reactions for the novel BHC fuels, was slightly diminished owing to the shorter combustion durations. Despite the reduced intensity of microexplosions, DNMBHC and DADMBHC droplets were found combust approximately 2.39 and 1.78 times faster than a similar-sized RP-1-s droplet with 80% ambient oxygen. Among the three novel BHC fuels, only DNMBHC fared better than RP-1-s in terms of maximum flame temperatures and had sooting propensities comparable to that of the benchmark fuel and thus was assessed as the most promising candidate.