Power Factor is the property of load. Generator is a source
of power. So it can not be corrected thru generator. However
Synchronous generators can regulate power factor by
adjusting it's excitation current.

Offcourse, p.f is the property of load. In a DG set, prime
mover supply mechanical BHP to the alternator, and it does
not depends on the load characterestics, but depends on the
load current.
in short, an alternator demands higher input torque/power
from the prime mover, as soon as current increases from
alternator.