We demonstrate a single-photon collection efficiency of (44.3 + or - 2.1)% from a quantum dot in a low-Q mode of a photonic-crystal cavity with a single-photon purity of g super((2)) (0) = (4 + or - 5)% recorded above the saturation power. The high efficiency is directly confirmed by detecting up to 962 + or - 46 kilocounts per second on a single-photon detector on another quantum dot coupled to the cavity mode. The high collection efficiency is found to be broadband, as is explained by detailed numerical simulations. Cavity-enhanced efficient excitation of quantum dots is obtained through phonon-mediated excitation and under these conditions, single-photon indistinguishability measurements reveal long coherence times reaching 0.77 + or - 0.19 ns in a weak-excitation regime. Our work demonstrates that photonic crystals provide a very promising platform for highly integrated generation of coherent single photons including the efficient out-coupling of the photons from the photonic chip.