No, you don't want to kill living flesh. Just wash with soapy water, salty water or hibiscrub (any of those will do) and rinse, dry with clean paper towels, pack with the cream and put the dressing on, then vetwrap.
ETA that plastic oral syringes (no needle) are great for squirting saline into...
If it's kernel, it won't bleed. It will be white or yellow, not pink fleshy material. A bit like cottage cheese, or it could be hard and cheddary. If you pack the wound with antibiotic cream that should kill off any remaining infection.