Should this have been bandaged by a professional?

Sliced through my fingernail into the flesh whilst peeling a carrot this morning, its taken about a cm of nail and flesh off with it.. I spoke to a nurse online, she said no need to go to minor injuries to get it treated I should do it myself and suggested I went to a pharmacy. I bandaged it but couldn’t clean it properly, the pain is extreme, it’s full on nerve pain. I put savlon on the bandage and wrapped it up. Slightly concerned about infection and can’t bare the thought of taking it off and cleaning it. How often should this be cleaned and re-applied? TIA