If you kill all the cells with the mutation, you get rid of the cancer. It can come back when you dont kill all the cancerous cells or another mutation happens. Cancerous cells are a lot more common than you think, but usually your immune system takes care of it.

Edit: important note its just some cells that mutate its not all the cells in your body.

You seem to be thinking of "the genetic mutation" as something separate from the cancerous cells. It's not.

That is, the genetic mutation that causes cancer isn't something that's expressed in your whole body that makes cancerous cells, it's something that happens in one cell that then divides uncontrollably (if the immune system doesn't eliminate is as foreign soon enough).

In general, one single mutation won’t give you cancer. You need some mutations on specific genes to develop it. Some of this mutations may come with the gene set you inherited from your parents, and all your body cells get this mutation. Other mutations develop during your lifetime by different reasons, but this mutations are not present on every cell your body have, but only on some of them. Everyday you get mutations on your DNA, but most of the cells that get these mutations get eliminated by the immune system. The problem is when you get the set of mutations needed for a cell to divide uncontrollably, and the immune system does not detect this. Cancer cells are more susceptibles to certain treatments that regular cells, so you can use these treatments to eliminate them. If you eliminate all cancer cells, you are cancer free, if not, you might get cancer again. People that inherit mutations from their parent genes, are also more susceptible to get cancer again, as they need fewer other mutations in their cells to develop it.

If all the cancer cells and precancerous cells (cells with some of the mutations needed to be cancerous but not all) are dead, you can get a new cancer from new mutations, but it's not more likely than before you got cancer. It's not the cancer coming back, it's a new one. Killing the precancerous cells is easier than the cancerous cells because they are localized around the main tumor. They can't go through metastasis and move away. They are killed when the main tumor is removed or destroyed with radiation.

Often not all cancer cell are dead, that's why the cancer comes back. Then it's the same cancer. Even if it appears in a different location, it's still the same one, it just got to the new place through metastasis.

The genes are inside every single cell. Those are the ones getting mutated. When they mutate our body will take care of it (hopefully) and get rid of them or the cell undergoes programmed cell death called apoptosis. If apoptosis fails and body cant clear it out, may have problems

You can have genetic mutations that leave you more susceptible to getting cancer. But it’s the cells that mutate…for example, a keratinocyte, that if dividing haphazardly will develop into squamous cell carcinoma (a skin cancer). You can’t change your DNA but things like lifestyle modifications go a long way. For example just because someone has a gene for breast cancer, does not mean that they will get breast cancer.

The answer is that the distribution of cells bearing the mutation(s) is spatially restricted, local to the organ that hosts the tumor. For example, a squamous cell carcinoma results from mutations in dermal skin cells, causing cancerous growth. The mutated cells are limited to the tumor itself. Hence, removal of the tumor will remove all of the mutated cells. Some cancers can become malignant and will spread to other organs. The mutated cells are defined by the original parent tumor and the sites of metastasis where those cells have migrated, elsewhere in the body.

Some cancers come back because whatever treatment - surgery, chemo, immunotherapy - has failed to eradicate all of the cells bearing the cancer-causing mutations.

There are some heritable cancers e.g. retinoblastoma, look up Al Knudson's two-hit hypothesis. Or the BRCA genes. In those cases, where all cells have the mutation, yes indeed cancer is hard to defeat. This is why Angelina Jolie had her breasts removed and replaced, she carried the BRCA allele.

In more common cancer types, cancer is the result of cumulative mutations over a lifetime. Sequences of mutations occurring one after the other, over years, until it reaches the cancer breaking point. So to cure these patients means removing those cells with those mutations, a subset of all cells. To not cure, means some cells remained. Cells can be destroyed with surgery (removal), radiation, chemotherapy, immunotherapy (which stimulates the immune system to destroy them, elegant)

Now, there is also the concept of field cancerization, if a tumor is removed, neighboring cells have SOME of the mutations seen in the cancer, just not enough to be themselves cancer, having to do with this lifelong process of sequential cancer mutations. This is a serious topic among cancer surgeons - how much should I remove? Should I remove extra for fear of field cancerization?

Keep in mind, the standard line is that all cells in our body have the exact same genome, in truth that is not the case (as cancer shows us) and recent articles in top journals in the past year have implicated that even our non-cancer cells carry slightly different genetic variation, thought to be mutations in the first month or two in utero. This was done by correlating mutations found only in specific tissue types with the embryonic origin of those tissues. As such we can say our cells our all 99.99999% the same DNA sequence as each other, but there are small differences and major subpopulations.

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You seem to misunderstand what a mutation means. A mutation is a random change in a gene. That happens millions of times in millions of cells in our body. Many of those mutations cause the cell to die, some don't change anything, some may make it get better at something, like not dying - putting it on the path to cancer.

If cell#74729194 mutates and becomes cancerous, it divides, divides and divides. If chemo poisons it and all its descendants, You should be cancer-free.

In reality, the more a cell's DNA is altered, the less stable it becomes, which makes every next generation of cancer different from the previous ones. That in turn makes it so all the cells are different, which can cause say 70% of the cells to respond to chemo while the rest won't.

The other problem You touched upon is what is considered a remission - yes, if we don't find a trace of it, we will consider it a remission - but we can't deconstruct a human to check every cell if there aren't 3 metastatic ones in that person's brain, one in the little left toe, and some next to where the tumor was. Best we can do is find a new small tumor, or some activity suggesting a metastasis, which can look similar to say an infection (and inflammation).

It is complicated, I think the first 2 paragraphs show what You wanted, and the rest is more corrections. Hopefully they are clear.