The Mishnah is about to discuss the idea of burning chametz on Erev Pesach, and particularly when the chametz is Terumah. That actually comes up in the next Mishnah, but in order to understand the basis for it we first need to discuss the idea of burning offerings: when one offering is definitely tamei, while regarding another offering there is no certainty whether it is tamei or not. Here are the details.
The words of Rabbi Chanina, the deputy Kohen Gadol:
The deputy of the Kohen Gadol said "Kol yemei ha-kehunah" - meaning, throughout the entire time that the kohanim served - they never refrained from burning meat of an offering that had become tamei through a "child of tumah." A "child of tumah" is not an av ha-tumah and is not the source of the tumah, but rather a secondary tumah, and the Gemara explains that this refers to sheni le-tumah.
To illustrate the order of the levels:
Av ha-tumah - a dead mouse.
Rishon le-tumah - an object that touched the dead mouse.
Sheni le-tumah (a child of tumah) - an object that touched the rishon.
Shelishi le-tumah - the meat of the offering that touched the sheni. This level exists with regard to Terumah and offerings, but not with regard to chullin and ordinary food.
The kohanim did not refrain from burning this meat of an offering, which had become tamei through contact with a sheni le-tumah, together with meat of an offering that had become tamei through an av ha-tumah, such as a dead mouse or a creeping creature. The meat that touched the creeping creature is a rishon le-tumah, and anything that touches it becomes a sheni le-tumah; and once the meat that had been a shelishi le-tumah is burned together with it, it rises to the level of sheni le-tumah, and the severity of its tumah increases.
Nevertheless, in the words of the Mishnah, "Af al pi shemosifin tumah al tumaso" - it is permitted, since both of them are disqualified in any case.
Rabbi Akiva's addition:
Rabbi Akiva added a further novel point about the practice in the days of the kohanim: throughout all the days of the kohanim they never refrained from burning oil that had been disqualified through a tevul yom. A "tevul yom" is someone who was tamei and immersed in a mikvah, but whose sun has not yet set - the sun has not yet gone down on the day of his purification. And the halachah is that a tevul yom, even though he has already immersed, has the status of a sheni le-tumah.
It emerges that the oil disqualified through contact with a sheni le-tumah is a shelishi le-tumah. And even so, they did not refrain from burning it in a lamp that had become tamei through a person who was tamei from a corpse - a lamp that became tamei through contact with a person who had touched a dead body. A person who touches a corpse is an av ha-tumah, and the lamp that touched him receives that same status.
The basis for this is a special halachah: "Cherev harei hu ke-chalal" - a metal utensil that touches an av ha-tumah itself becomes an av ha-tumah. Therefore the metal lamp has the status of an av ha-tumah, and the oil burned in it becomes a rishon le-tumah.
It turns out that here there is a jump of an additional level: according to Rabbi Chanina, the deputy Kohen Gadol, we take something that was a shelishi le-tumah and make it a sheni le-tumah; whereas here the oil, which was a shelishi le-tumah because of contact with a tevul yom, becomes a rishon le-tumah because it is burned in a lamp that has the status of an av ha-tumah.
And nevertheless, said Rabbi Akiva, this is what they would do - "Af al pi shemosifin tumah al tumaso" - even though they add tumah onto its existing tumah.