Will Amoksiklav help with a burning sensation in the head during a cold?

It is difficult to assess the cause of the ailment you have described. The antibiotic Amoksiklav has an antibacterial effect and is used to treat, m.in, bacterial infections of the upper and lower respiratory tract (sinuses, ear, bronchi, lungs). If the "burning" is related to the course of the infection, the antibiotic should alleviate the symptoms.

Burning sensations in the throat can be soothed with anesthetics, e.g.:

https://www.gdziepolek.pl/produkty/57942/orofar-max-pastylki-lamelki-zelki/apteki

https://www.gdziepolek.pl/produkty/75015/strepsils-intensive-tabletki-do-ssania-gryzienia-zucia/apteki

You can also use lozenges or aerosol moisturizers:

https://www.gdziepolek.pl/produkty/94688/gardvit-olive-aerozol/apteki

https://www.gdziepolek.pl/produkty/94694/cevitt-gardlo-tabletki-do-ssania-gryzienia-zucia/apteki

If you feel discomfort in the nose, you can use inhalations and use mucosal moisturizers:

https://www.gdziepolek.pl/produkty/67588/rinopanteina-masc-do-nosa/apteki

https://www.gdziepolek.pl/produkty/54808/nozoil-aerozol/apteki

During antibiotic treatment, it is worth using probiotics, which have a protective effect on the gastrointestinal tract and allow you to avoid intestinal side effects.

I recommend the article:

https://www.gdziepolek.pl/artykuly/fakty-i-mity-o-antybiotykach