  We demonstrate a single-photon collection efficiency of $(44.3\pm2.1)\%$ from
a quantum dot in a low-Q mode of a photonic-crystal cavity with a single-photon
purity of $g^{(2)}(0)=(4\pm5)\%$ recorded above the saturation power. The high
efficiency is directly confirmed by detecting up to $962\pm46$ 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 $0.77\pm0.19$ 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.
