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Dark counts in superconducting single-photon detectors (SSPDs) manifest themselves as spontaneous, transient voltage pulses, typically indistinguishable from photon counts. We present here a new readout technique based on integrating the SSPD with a low-noise, cryogenic high-electron-mobility transistor (HEMT) with high-input impedance. This arrangement allowed us to achieve amplitude resolution of the recorded output transients. In two-dimensional superconducting nanostripes, the physics of photon counting is based on the hotspot formation mechanism, while the dark counts correspond to voltage transients triggered by the vortex-antivortex motion and/or phase-slip centers. Thus, their respective transients can be distinguished by comparing the output pulse amplitude distributions. Our scheme also allowed us to perform photon-energy-resolution studies by comparing the SSPD output pulse amplitude distributions (the mean pulse amplitude and the distribution width) collected for incident single photons with different energies.