As mentioned on daf 36, we have two possible ways to interpret our mishna. The divergence is based on the dispute between chananya and the rabbis about שהייה- leaving food from before Shabbos on exposed fire to cook by itself. The rabbis forbid this (bc one may be tempted to stoke the coals) while chananya permits it if the food is already מאכל בן דרוסאי- jsut edible. The gemara is ultimately unsure how our tanna here holds.

On Shabbos, beis Hillel permits חזרה- returning food to a fire, as long as the fire is גרוף וקטור- coals clearer away or ash covers the coals. However, this is qualified only to return food to the top of a stovetop but not to return the food inside the actual stove. (Looks too much like cooking)

As we’ve learned, one can’t put uncooked food on exposed coals to cook on Shabbos. (Like chananya, this is only b4 it’s edible) This law only restricts one to put food inside or on top of the Stovetop, but it is permitted to put it next to the stovetop. (לסמוך)

According to the rabanan that restrict שהייה for any uncooked food, there is a dispute in the genara whst they hold about מצתמק ויפה לו- fully cooked food that gets better the longer it will stay on the fire. If staying on the fire will be bad for the food, it certainly is ok(like flour based foods, figs, cereals, or dates.)

If someone placed ash over the coals to make it קטומה, but then the coals flared up again by themselves, the gemara is unsure if שהייה is allowed. Perhaps it is ok bc an act of cooling the coals was performed or perhaps is is assur bc now the coals are strong again. The gemara seems to also suggest a compromise, where uncooked food will be forbidden but fully cooked food that’s מצתמק ויפה לו will be permitted.