if the specific band is quite strong then the smear will just produce faint random sequences with the primer incorporated in the sequences and will just show as a slightly higher background to the main sequence . I would sequence the product and expect it to work
If you are referring to the primer dimers, then it's better to do a cleanup to remove them. They can add too much background noise to your reaction. Or you may have a smear from loading too much PCR product. Worst case scenario is that you have amplified more than one product. In that case, then you will get a LOT of noise and the you won't get good sequence data. Share a picture of your gel and I can be more specific.
From the gel image, it might just be primer dimers on the low end, plus you may also have too much PCR product loaded on gel. As the top end of your gel, the smear is also visible, you only need a few ng of PCR for sequence reaction, you can still get good sequence back.
However, if you can purify the target band of the main PCR band it will avoid the possible noises , and you will get much better sequence.
Yes, it looks good. Just cut out the band and do a 'gel extraction' purification (commercial kit available). Cut the band as quick as you can to avoid long exposure to UV. UV can damage DNA. Then, subject it to sequencing.