When the suppository is inside you do not feel discomfort because the pH (acidity) of the product is the right pH inside.
However, when the melted liquid leaks out, it may cause a burning sensation to the outside skin.
The normal pH inside the vagina is mildly acidic like pH 4.0, but the pH outside is neutral like pH 7.0.
The skin is used to the neutral pH but it may not tolerate acidic pH. To reduce discomfort, you can wash it to remove the leaked liquid. The discomfort will go away.
Next time, you can place the suppository deeper inside and use it at bedtime. This will allow a whole night to absorb the product. There will be less unabsorbed material leaking out the second day or leaking out inconspicuously a little at a time without causing too much discomfort. You may wear a pantiliner for protection on the second day only. The 3rd will be largely clean.
This is like a regurgitation of gastric juice into the oral cavity. You will feel very acidic. This is normal because the stomach fluid is acidic. If regurgitation happens, you just rinse your mouth and the discomfort will go away.
Mild discomfort is a part of the treatment. It will go away soon. Do not worry too much. It will be fine.