Oh no! The Character has an ouchie! And they can't go to urgent care to have it looked at because that's not invented yet! Whatever shall we do!!!
Keep. It. Clean. Carefully and gently clean out gross stuff and things that do not belong in there, do not introduce foreign matter, and keep that wound covered.
The body will support Caretaker in this. Exudation is a natural defense mechanism to flush out germs and other dirt. Normal exudate is clear, with a slight yellow tinge, and almost odorless (wounds do have a characteristic smell, but they don't stink). Excessive exudate can form crusts around the wound, those should be cleaned away. Boiled water or saline solution is probably the safest bet for wound irrigation. Gentle mechanical cleaning (wiping with wet cloth) is appropriate.
Wounds want to be kept moist. Ointment can help with that, and also prevent the bandage from sticking to the wound. For a premodern setting, lard (rendered pig fat) is probably not a terrible option. It contains almost no water or sugar, so doesn't grow mold, and the grease prevents the tissue from drying out. Still, Caretaker will probably want to soak that bandage before they remove it.
The bandage should be clean linen. Not wool. If Caretaker has to rip up their clothes for bandages, at least use the undergarment, and the parts of it that are less dirty (sleeves).
A cavernous wound (ie from a drained abscess) might need to be tamponaded, that is, lined with a strip of wet cloth, to keep it open and help with drainage.
Various herbs, honey, spices, etc, have historically all been used to promote healing, some more successfully than others. However, everything is potentially contaminated with germs and fungal spores. Putting mashed plant matter that hasn't been otherwise treated into wounds is always a gamble.
Disinfection is a possibility, but an effective disinfectant will also damage healthy tissue. If Character has their wound bathed with wine or spirits every time the bandage is changed, that wound will become necrotic.
Stitches are likewise a double-edged sword. Pulling the edges of a deep wound together will make it heal quicker and more smoothly, but it also retains exudate in a warm, anaerobic environment, which makes an ideal growing environment for nasty bacteria. Balance risk and reward appropriately.
Some additional thoughts:
Medicine has taken unimaginable leaps during the last ~150 years. Things that would have been a death sentence to our ancestors are barely more than a nuisance to us. As a result, we have a fundamentally different attitude towards treatments, especially invasive treatments. We do surgeries as precaution or to restore optimal function, where in the past, surgery was often a last resort. It was very risky, and ran a good chance of killing the patient, so it would only be done if they would die anyway.
That doesn't mean doctors and surgeons of the past were helpless, there are a lot of amazing and very sophisticated treatments, that are older than you'd think. For as long as we have been human we have tried to heal each other, and were often successful.
Just because we know humoral pathology is wrong doesn't mean it's stupid. It's based on very solid observations of health and the human body. It was the accepted theory for a long time. It makes no sense for a doctor in that setting to reject it.
The human body is at once more resilient and more fragile than most people think. Humans can walk off devastating injuries, a lot of things will sort themselves out. However, once a tipping point is reached and the body can no longer compensate for whatever is going wrong, there's no turning that process around. This is why systemic infections are so dangerous: Bacteria/toxins overwhelm the immune system, everything is poisoned, multiple organ failure, death. It's very difficult to come back from that, even today.
THIS IS WRITING ADVICE, NOT MEDICAL ADVICE. In real life, there's no benevolent writer that makes sure you get the best possible outcome. Real illnesses don't care how relevant you are to the plot. Have that looked at by a professional.
