When a low-power finite-energy Airy pulse (FEAP) and a high-power soliton pulse simultaneously propagate in an optical fiber, we numerically study the evolution of the FEAP affected by higher-order effects, including third-order dispersion (TOD), Raman scattering and self-steepening (SS). It is found that shedding solitons are generated from the FEAP due to the effect of cross-phase modulation (XPM). The TOD only affects the center position of the shedding soliton, but does not change the spectrum structure. The truncation coefficient of a FEAP, TOD and SS can be used to manipulate the Raman-i...