I researched a bit at:

wikipedia.com under hand washing & I checked out neosporin at
drugs.com & wikipedia.com

Hand washing without soap flushes larger particles away from the injured area. Soap & water does a better job of removing oils & finer particles that would stick to the injured area. Soap & warm-hot water accelerates the process (but doesn't kill germs.) Antibiotic soap & warm-hot water cleans like regular soap & water but also kill many germs & temporarily prevents them from returning.

Neosporin is a combination of 3 antibiotics (bacitracin, neomycin, and polymyxin B) & a mild pain killer. The 3 antibiotics kill a wider range of germs. They are intended for topical cuts & breaks in the skin. More powerful antibiotics are mostly unneccesary at this time, & could have side effects or the patient could have undiagnosed allergies.

I find that creams can collect more dirt & grime after an application so a bandaid would be needed (especially if you were outside & continuing to expose yourself to dirt, sweat etc.) For me bandaids keep a scab from forming & closing the wound. I will often take off a bandaid at night to let it weep, air out & form a scab. Naturally, I don't want it to be completely untreated so I would use povidone iodine. I used to use rubbing alcohol or peroxide, but I heard that those damage the skin. Povidone iodine can stain cloth so I often use Hibiclens which goes on like weak cough syrup & doesn't stain as bad (or at all.) For my first aid kit I keep povidone iodine & Hibiclens in small plastic weapons oil bottles that I bought at the army surplus store. The bottles are opaque as a peroxide bottle so I don't worry about light sensitive liquid inside. The bottle tops open like an Elmers Glue bottle so I don't worry about them leaking.

Hope that helps.

Tim