  We propose a simple quantum-key-distribution (QKD) scheme for practical
single photon sources (SPSs), which works even with a moderate suppression of
the second-order correlation $g^{(2)}$ of the source. The scheme utilizes a
passive preparation of a decoy state by monitoring a fraction of the signal via
an additional beam splitter and a detector at the sender's side to monitor
photon number splitting attacks. We show that the achievable distance increases
with the precision with which the sub-Poissonian tendency is confirmed in
higher photon number distribution of the source, rather than with actual
suppression of the multi-photon emission events. We present an example of the
secure key generation rate in the case of a poor SPS with $g^{(2)} = 0.19$, in
which no secure key is produced with the conventional QKD scheme, and show that
learning the photon-number distribution up to several numbers is sufficient for
achieving almost the same achievable distance as that of an ideal SPS.
