If you want to employ one of the PCR primers in the sequencing
reaction you must elute the product of interest from an agarose
gel. Otherwise, a mixed sequence will be produced as the primer is
most likely to bind both the specific and unspecific products. If
you employ a specific nested primer in the sequencing reaction, it
might work. However, producing only one product in the PCR reaction
is the best guarantee for a good sequence. In almost all cases,
improving the PCR conditions (specific primer design, higher
annealing temperature and/or lower primer concentration) helps to
avoid unspecific products.