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.