Penicillin is an anti-biotic that does absolutely nothing to fungus, viruses or protozoan infections.
Anti-biotics should only be used on known bacterial infections that have not responded to normal treatments. Improper use and mis-use of anti-biotics has lead to drug resistant bacteria that kill people, birds, fish, animals and people all over the world.
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The fish doesn't appear to have a bacterial or fungal infection on the head. It looks fine to me. If you can provide a different picture or video showing the issue, it might help, but at this stage it seems fine.
If the fish has a white film over its body or fins, check the water quality for ammonia, nitrite, nitrate and pH, and do a big water change and gravel clean the substrate. Clean the filter if it hasn't been done in the last few weeks.
If you have livebearers that get sick, the best treatment is clean water and salt.