We present a new operation regime of NbN superconducting single-photon detectors (SSPDs) by integrating them with a low-noise cryogenic high-electron-mobility transistor and a high-load resistor. The integrated sensors are designed to get a better understanding of the origin of dark counts triggered by the detector, as our scheme allows us to distinguish the origin of dark pulses from the actual photon pulses in SSPDs. The presented approach is based on a statistical analysis of amplitude distributions of recorded trains of the SSPD photoresponse transients. It also enables to obtain information on energy of the incident photons, as well as demonstrates some photon-number-resolving capability of meander-type SSPDs.