A semiconductor quantum dot can generate highly coherent and indistinguishable single photons. However, intrinsic semiconductor dephasing mechanisms can reduce the visibility of two-photon interference. For an electron in a quantum dot, a fundamental dephasing process is the hyperfine interaction with the nuclear spin bath. Here, we directly probe the consequence of the fluctuating nuclear spins on the elastic and inelastic scattered photon spectra from a resident electron in a single dot. We find the in-plane component of the nuclear Overhauser field leads to detuned Raman scattered photons, broadened over experimental time scales by field fluctuations, which are distinguishable from both the elastic and incoherent components of the resonance fluorescence. This significantly reduces two-photon interference visibility. However, we demonstrate successful screening of the nuclear spin noise, which enables the generation of coherent single photons that exhibit high visibility two-photon interference. DOI: 10.1103/PhysRevLett.116.257401 Indistinguishable single photons are an essential resource for quantum photonic logic gates and networking [1]. Among the various approaches for generating identical light quanta, resonance fluorescence (RF) from a semiconductor quantum dot (QD) [2-10] is one of the most promising for practical technological implementation. The RF spectrum is composed of elastic and inelastic scattered light [11][12][13][14]. The elastic or Rayleigh scattered light, first measured in homodyne absorption experiments on a QD [15], has the first-order coherence properties of the laser but the second-order coherence properties of the emitter [16]. This light, which can be imprinted with an arbitrary phase or temporal profile [3], is fundamentally indistinguishable. RF can therefore relax the requirement for a perfectly stable, transform-limited optical transition that is difficult to realize in the solid state. Specifically, compared to nonresonant excitation followed by spontaneous emission, RF helps overcome the relatively slow spectral fluctuations caused by charge noise in the QD environment [14,17].In addition to the elastic and incoherent components of RF, near resonant excitation can lead to Raman scattering into another ground state. Taken in isolation, Raman scattering from a QD with a resident electron or hole can have several attractive features. First, below saturation, the coherence of the Raman photons is determined by the ground-state dephasing rather than that of the excited state [18][19][20].Second, Raman photons are highly tunable: their energy is determined by the detuning of the driving field rather than the fundamental optical transition energy [18,20,21]. Finally, because of clean selection rules in QDs, the polarization of the Raman scattered photons can be linked to the spin of the final state [20][21][22], enabling spin-photon entanglement [23][24][25] and raising the prospect for quantum networks [26]. Typically, spin-flip Raman photons are generated using an in-plane extern...