  Because of their universal nature, Fano fluctuations are expected to
influence the response of superconducting nanowire single-photon detectors
(SNSPDs). We predict that photon counting rate ($PCR$) as a function of bias
current ($I_B$) in SNSPDs is described by an integral over a transverse
coordinate-dependent complementary error function. The latter describes
smearing of local responses due to Fano fluctuations of the amount of energy
deposited into electronic system. The finite width, $\sigma$, of the $PCR$ vs
$I_B$ arises from fluctuations in the energy partition between quasiparticles
and phonons during the energy down-conversion cascade. In narrow-nanowire
SNSPDs the local responses are uniform, and the effect of Fano-fluctuations on
$\sigma$ is dominant. In wide-nanowire SNSPDs with strong coordinate dependence
of local responses due to vortex-antivortex unbinding and vortex entry from
edges, Fano-fluctuations smear singularities imprinted by vorticity on the
transition part of $PCR$ curve. We demonstrate good agreement between theory
and experiments for a series of bath temperatures and photon energies in
narrow-wire WSi SNSPDs. The time-resolved hotspot relaxation curves predicted
by Fano fluctuations match the Lorentzian shapes observed in experiments over
the whole range of bias currents investigated except for their tails.
