It kind of depends what you mean by "target". The immune system can certainly recognize elements inside the cell, and determine that they are foreign and need to be destroyed. The immune system can't really specifically remove those foreign elements, though (though there have been arguments that under some conditions it can). In general if a cell contains harmful foreign elements, the immune system rapidly identifies it and destroys the whole cell.

You ask about mitochondria specifically. In fact mitochondria are strongly recognized by the immune system and treated as harmful. This shouldn't be surprising in principle, because of course mitochondria are symbiotic bacteria, and the immune system is tuned to recognize and destroy bacteria. Normally, though, mitochondria are invisible to immunity because they are intracellular; it's mainly after cells are damaged that mitochondria are exposed to the immune system and can lead to responses.

>Mitochondrial damage-associated molecular patterns (DAMPs) are molecules that are released from mitochondria to extracellular space during cell death and include not only proteins but also DNA or lipids. Mitochondrial DAMPs induce inflammatory responses and are critically involved in the pathogenesis of various diseases. ..

--The Roles of Mitochondrial Damage-Associated Molecular Patterns in Diseases

There are two major pathways by which cells provide information about their internal components, to the cellular components of the immune system like lymphocytes and neutrophils.

In one, there are a large number of intracellular sensors that monitor cells for general pathogen-associated molecular patterns (wikipedia link). These trigger pathways that eventually result in the cell producing cytokines, like interferons, that activate and recruit immune responses.

In another branch, there's a complex process that constantly surveys intracellular protein production, and moves samples to the outside of the cell where lymphocytes can analyze the cell and respond to those that have abnormal components. This is called antigen presentation.

But again, the response is not so subtle. Instead of delicately removing the abnormal component, the whole cell is typically destroyed, presumably preventing the intracellular pathogen from completing its life cycle and amplifying its numbers.

Interferons and interleukins are two good examples of it.

Interferons can be produced by infected cells or immune cells in response to infections or tumors. One of their functions is to slow down replication of RNA to slow down the replication of the pathogen or the cancer cells. And interferons also targets other cells to lower their vulnerability against the infection.

Interleukins (or cytokines) are mostly regulators of the immune response. They activate and target immune cells against pathogens or antigens, they control their replication, the production of antibodies and the differentiation of the many immune cells (like memory lymphocytes, useful against a second infection). Cytokines are produced by immune cells but also many other cells in your body and another of their functions is to regulate the immune response to avoid self-immune reactions. A cytokine storm is an acute disproportionate cytokinic response to an infection. That's what caused so many deaths at the begining of the COVID pandemy.

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The main way the immune cells recognise foreign things inside a cell is by detecting them outside the cell. Part of a cell's natural processes is to breakdown some components that are inside the cell and 'present' them on the outside of the cell. These are called antigens, or peptides.

If they are 'self' peptides, then the immune system has been trained to recognise these as safe and so it doesn't react to them.

If they are 'foreign' peptides, then the immune system recognises them, and subsequently targets that cell for destruction - either by releasing chemicals that perforate the cell membrane, or by recruiting other cells to do the job.

You can get some treatments, such as antibodies, which can have a chemical drug attached (ADC - antibody-drug conjugate). The antibody targets a specific cell type and then when it gets there, it releases the drug, which goes into the cell and targets a disease, or kills the cell itself.

There are other methods, some described in other answers that work in different ways but have similar aims of targeting something inside the cell.

When things go wrong with this: The immune system incorrectly recognises self peptide as foreign and attacks healthy cells (autoimmune disease) The immune system does not recognize foreign peptide correctly The disease stops the infected cell from presenting the foreign peptide on the cell surface - avoiding detection The disease hijacks the presentation process and gets the cell to present a self peptide that would normally not be presented suppressing the immune response that would normally happen.