We demonstrate and evaluate an on-demand source of single itinerant microwave photons. Photons are generated using a highly coherent, fixed-frequency qubit-cavity system, and a protocol where the microwave control field is far detuned from the photon emission frequency. By using a Josephson parametric amplifier (JPA), we perform efficient single-quadrature detection of the state emerging from the cavity. We characterize the imperfections of the photon generation and detection, including detection inefficiency and state infidelity caused by measurement backaction over a range of JPA gains from 17 to 33 dB. We observe that both detection efficiency and undesirable backaction increase with JPA gain. We find that the density matrix has its maximum single photon component $rho_{11} = 0.36 pm 0.01$ at 29 dB JPA gain. At this gain, backaction of the JPA creates cavity photon number fluctuations that we model as a thermal distribution with an average photon number $bar{n} = 0.041 pm 0.003$.
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