Menachos chapter 3, mishnah 3. This mishnah too deals with the laws of mixtures in menachos, and it presents three cases.
The first case - "shtei menachos shelo nikmetzu" - two minchah offerings from which the kometz had not yet been taken:
Before us are two quantities belonging to two separate menachos, minchah A and minchah B, which have come into the same vessel - but before the kemitzah. A kometz must be taken from each one separately, for the verse says "vekamatz mishom melo kumtzo misalteha" - from its own fine flour and not from that of another. Therefore we require a pure kometz from minchah A and a pure kometz from minchah B, without any component of the second minchah becoming mixed into the kometz.
From here the matter becomes a practical question: "im yachol likmotz mizo bifnei atzmah umizo bifnei atzmah" - if he can take a kometz from this one on its own and from that one on its own - for example, minchah A fell on the left side of the vessel and minchah B on the right side, and they do not completely overlap one another, so that from the two extremes a pure kometz can be obtained from each of them - "kesheiros" - they are valid. The two pure kematzim are burned on the mizbeiach and all is in order. "Ve'im lav - pesulos" - and if not, they are invalid, and there is no second-best option here: any kometz that is mixed, that was not taken entirely from one minchah, is invalid.
The second case - "hakometz shenisarev beminchah shelo nikmetzah" - a kometz that became mixed into a minchah from which the kometz had not been taken:
A kometz from minchah A became mixed into minchah B, from which the kometz had not been taken. Only the kometz is supposed to go up on the mizbeiach, and one cannot simply pour everything onto the fire: since no kemitzah was performed on minchah B, the bulk of that minchah is not designated to go up on the mizbeiach but to be eaten. And since it is improper to burn on the mizbeiach that which is not designated to be burned upon it, there is a prohibition here, a rabbinic prohibition, in the very act of placing the whole mixture on the fire. Therefore "lo yaktir" - he may not burn it, and if the mixture cannot be separated and the kometz cannot be extracted without offering up everything, he must not do so.
"Ve'im hiktir" - and if he did burn it, that is, if he transgressed the prohibition and placed the entire mixture on the mizbeiach: "zo shenikmetzah alsah labe'alim" - the one whose kometz was taken counts for its owners, since its kometz was burned and after the fact they have fulfilled their obligation; whereas "zo shelo nikmetzah lo alsah labe'alim" - the one whose kometz was not taken does not count for its owners.
One might have mistakenly thought that just as a kohen's minchah is burned in its entirety with no kemitzah at all, so too a Yisrael's minchah could be burned in its entirety after the fact without any specific kometz being taken from it, and it would be valid bedieved. This is not so: a Yisrael's minchah before the kemitzah is not fit to go up; it requires kemitzah, and only the kometz goes up on the mizbeiach. After the fact, if the kometz went up together with other things, those other things do not invalidate the pure kometz; but in our case, since no kemitzah at all was performed on the second minchah, that minchah does not count at all, while the first is valid after the fact, even though something was done to it that should not have been done.
The third case - "nisarev kumtzah bishyareha o bishyareha shel chavertah" - its kometz became mixed with its own remainder or with the remainder of another minchah:
The kometz becomes mixed with the remainder - either with the remainder of that very same minchah, which had been separated into kometz and remainder and then became mixed again, or with the remainder of minchah B. Here too the remainder has no business going up on the mizbeiach, but unlike the previous case, here we are speaking of a defined remainder and not of a minchah before the kemitzah, and there is a Torah law, based on verses, that the remainder does not go up on the mizbeiach. It emerges that offering it up on the mizbeiach here involves a Torah prohibition.
Even so, "ve'im hiktir - alah labe'alim" - and if he burned it, it counts for its owners: although offering the remainder up on the mizbeiach is forbidden by Torah law, the pure kometz went up and was burned, and therefore the owner has fulfilled his obligation after the fact.
The disqualifications of tamei and yotzei:
From here the Mishnah opens a new topic (and in the Babylonian Talmud this is even counted as a new Mishnah). Once the kemitzah has been performed, two general things can go wrong with that handful, and these are the disqualifications: tamei (impure) and yotzei (taken outside). The handful may become impure through contact with a source of tumah, or it may be taken out of the Azarah even for a moment. Anything that is consecrated generally has a defined area to which it belongs, and the moment it leaves that area it becomes permanently disqualified. The handful must remain in the Azarah, and once it has left it even momentarily it is too late, and it can no longer be brought up on the altar. These two disqualifications, then, invalidate the handful.
But after the fact, if they were brought up on the altar, has the owner fulfilled his obligation? That depends. There is the tzitz, the gold plate that sits on the forehead of the Kohen Gadol, one of his eight garments, engraved with "Kodesh LaHashem," and it effects acceptance: it causes certain disqualified offerings to become acceptable. The verse itself does not specify which type is meant, but says: "V'hayah al metzach Aharon" - the tzitz shall be on the forehead of the Kohen Gadol, "v'nasa Aharon es avon hakodashim" - and through it certain iniquities of the sacred offerings will be borne away, "v'hayah al mitzcho tamid l'ratzon lahem lifnei Hashem" - the tzitz shall be on his forehead always and shall effect acceptance before Hashem. But it is not stated what exactly it comes to permit.
With regard to tumah there is a significant leniency: certain offerings were permitted to be brought even in a state of impurity, such as communal impurity at the appointed time and the like. It emerges that tumah is a less severe restriction, and therefore it is the offering or the meal offering that became impure that is rendered acceptable through the tzitz, and not one that was taken outside.
Therefore the Mishnah says as follows: if the offering became impure and was nevertheless brought on the altar, whether the prohibition was transgressed deliberately or whether it happened inadvertently, the tzitz on the forehead of the Kohen, under the appropriate circumstances, renders it valid and accepted after the fact. But if the offering left the Azarah even for a moment, it becomes disqualified and the owner has not fulfilled his obligation, and the tzitz is of no help to it at all. The tzitz permits only certain categories of flawed offerings - chief among them those that became impure - but not one that was taken outside.
There is reason to emphasize this point again, since one might have reasoned that yotzei is the lighter of the two and therefore deserves to be included in the acceptance effected by the tzitz. What is the leniency in yotzei relative to tamei? During a period when private altars were permitted - such as when the Mishkan stood in Nov and afterward in Givon, throughout those fifty-seven years - a person was allowed to build an altar in his courtyard and bring offerings upon it. And a private altar has no curtains or partitions surrounding it, and therefore there is no prohibition of yotzei there during that period. It emerges that the disqualification of yotzei does not apply to offerings in every situation, and hence the inclination to treat it with some measure of leniency.
Bottom line: the answer is no. The tzitz accomplishes its effect only with regard to things that were impure and ultimately went up on the altar, but not with regard to things disqualified by the disqualification of yotzei.