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Heralded single-photon sources (HSPS) intrinsically suffer from multiphoton emission, leading to a trade-off between the source's quality and the heralding rate. A solution to this problem is to use photon-number-resolving (PNR) detectors to filter out the heralding events where more than one photon pair is created. Here, we demonstrate the use of a high-efficiency PNR superconducting nanowire single-photon detector (SNSPD) as a heralding detector for a HSPS. By filtering out higher-order heralding detections, we can reduce the $g^{(2)}(0)$ of the heralded single photon by $(26.6 \pm 0.2)\,\%$, or alternatively, for a fixed pump power, increasing the heralding rate by a factor of $1.363 \pm 0.004$ for a fixed $g^{(2)}(0)$. Additionally, we use the detector to directly measure the photon-number distribution of a thermal mode and calculate the unheralded $g^{(2)}(0)$. We show the possibility to perform $g^{(2)}(0)$ measurements with only one PNR detector, with the results in agreement with those obtained by more common-place techniques which use multiple threshold detectors. Our work shows that efficient PNR SNSPDs can significantly improve the performance of HSPSs and can precisely characterize them, making these detectors a useful tool for a wide range of optical quantum information protocols.