My answer to this question is, with caveats, no.
These are my personal views, written at three something a.m. with blurry eyes and a fogged brain, but I have been putting off answering this but feel further comment is needed.
Let us take "psychosomatic". It literally implies a connection between mind and body. That is where the problems begin.
Exploring the connection between mind and body is a valid discipline. Its not science though. Its metaphysics.
We have to backtrack to what is mind. Classically its some thing, some entity, separate from body, but connected to body somehow. There are lots of sophisticated arguments as to why this is so, but the simple answer, and in my view much more likely to be correct, is there is no mind in this sense. Mind is just what we observe of brain function. What we call mind arises out of the activity of the brain.
Psychogenic is something else again. A psychogenic illness would be one in which the mind, the psyche, causes the illness. In the case of physical illness there would have to be a concrete mechanism connecting mind to the many physiological abnormalities we find in these patients ... like for example in IBS or ME, though in past times we can include claims made about so many diseases its almost universal, including heart disease and cancer. No such mechanisms have been provided. If such mechanisms are ever provided then most likely they would connect brain and disease, not mind and disease.
So in its literal sense, there is no evidence that any psychosomatic illness exists. Taken in a broader sense, in which what we call mind is a property of brain, a emergent function, its a whole other issue. Brain disease clearly exists. So this might fall under that category. However this creates a further caveat on mental disease.
Mental disease is a misnomer. Mental disease is so broad a category, and so badly founded in science, that once its framed as brain disease there are problems. Some clearly would be brain diseases, and require better understanding of brain, and how it goes wrong, to find cures. Yes, cures. If there are no cures, then we don't really have a proper understanding. So you should be able to go to your doctor, be objectively diagnosed, and given a cure, and you don't see them again.
On the other hand, some of what a few might want to call brain disease are just where normal operation of brain is askew ... the brain has learned the wrong things. These are more properly psychological disorders, not a disease, and in particular not a disease of the brain. Its also socially dangerous to define some of these things as diseases, because that would cover things from cult brainwashing to being loyal to a local sports team, or your favourite celebrity. I would not want to call those diseases.
If you allow psychosomatic disease to include brain disease, and psychological disorders, and not to include psychogenic disease, then yes, psychosomatic illness exists. Otherwise, no.
Psychosomatic, and worse, psychogenic, imply a range of hypothetical ideas that are unproven and may be just wrong. When you start from a basis of psychosomatic and psychogenic, and take these ideas seriously, then you run the risk of making spurious inferences, and then claims based upon those inferences. In my view its pseudoscience.
Psychiatry needs enough research development so that patients can be treated for brain disease, which is neurology, or patients can be treated for psychological disorders, which is psychology. There will also be some who have both of course, which is either comorbidity or a hybrid discipline psychoneurology or neuropsychology, though psychiatry might in the end find a home there. In my view psychiatry exists right now because we still don't understand brain. It has trappings of science, but is largely unscientific, and hence mostly pseudoscience.
I do think that will change, but I think its still a long road ahead. Objective diagnostic tests are a critical step in getting there. Just like they are for ME or any disease or disorder that is not understood.