There’s no home method for this beyond gentle cleaning, and knowing that is the most useful thing on this page.
Cleaning should be gentle. Soak a gauze pad or soft cloth in saline or mild soapy water and gently dab or wipe the skin, trying to remove drainage and any dried blood or other matter built up on it. With a closed surgical wound, do not rub or scrub the wound. Picking, digging, or working at material that won’t lift is not part of wound cleaning.
The one method for washing material out of a wound is something a clinician has to approve. Your provider may also ask you to irrigate, or wash out, your wound: fill a syringe with salt water or soapy water, whichever your provider recommends, hold the syringe 1 to 6 inches away from the wound, and spray hard enough into the wound to wash away drainage and discharge. That is a technique taught for a particular wound, not a general instruction.
There’s also a reason not to leave the problem alone. Having a foreign body, such as a piece of wood or broken glass, in the wound is one of the reasons wounds fail to heal. Material that stays in a wound can keep it from closing.
A clinician deals with this directly. During a debridement of a foot ulcer, the wound is probed with a metal instrument to see how deep it is and to see if there is any foreign material or an object in the ulcer. Finding and removing what is in a wound is a procedure. You should never try to do debridement yourself.
So for dirt that won’t come out: clean gently, irrigate only if your provider directs it, and know that material left in a wound is a reason wounds don’t heal. Removing it is done at an exam. Anything beyond gentle cleaning is a reason to be seen, not a technique to step up at home.