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We present an analysis of the performance of a broadband secondary radiation source based on a high-gradient laser-plasma wakefield electron accelerator. In more detail, we report studies of compact and ultra-short X-ray generation via betatron oscillations in plasma channels. For the specific working point examined in this paper, determined by the needs of other experiments ongoing at the facility, at ∼0.02 Hz operation rate, we have found ≲<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mn>6</mn></msup></semantics></math></inline-formula> photons emitted per shot (with a fluctuation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>50</mn><mo>%</mo></mrow></semantics></math></inline-formula>) in the soft X-rays, corresponding to a critical energy of ∼0.8 keV (with a fluctuation of 40%). The source will be implemented for experiments in time-domain spectroscopy, e.g., biological specimens, and for other applications oriented to medical physics.