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.