The majority of photonic quantum information technologies rely on single photons that have high purity and indistinguishability. Although solid-state quantum emitters can serve such single photons on demand, their asymmetric temporal and spatial mode profiles limit the optimal efficiency and fidelity of quantum interaction. Here, we demonstrate single-photon pulses at a telecom wavelength with a Gaussian-like temporal mode profile from a cavity-coupled single quantum dot. Engineering the exciton dynamics via multi-exciton cascade recombination and cavity detuning enables us to modify the rise and decay dynamics of single excitons. Furthermore, the cascade recombination process temporally retards the single-exciton emission from the background emission, leading to possible purification of single photons at high excitation power. In addition, coupling quantum dots into a low Q cavity mode leads to a Gaussian-like spatial mode profile, which brings a high collection efficiency. This approach paves the way for producing single photons with an optimized temporal and spatial waveform.