As with all other first- and second-declension nouns in both Latin and Greek, ψυχή has two distinct standard compounding forms in Ancient Greek (and a host of nonstandard ones that need not bother us here):
- ψυχο- is used if the second member of the compound starts with a consonant
- ψυχ- is used if the second member of the compound starts with a vowel
As such, your point is completely valid that, if psychoanalysis had been a word formed in Ancient Greek, it would (probably) have been ψυχανάλυσις.
As is mentioned in the comments, however, the word was formed in German, by Sigmund Freud, from two German elements: the combining form psycho-, and the Germanised word Analyse. Earlier on, he had used psychische Analyse.
It has generally been the case with such combinations involving old first- and second-declension nouns from Latin and Greek that the distinction between the prevocalic and the preconsonantal forms has become blurred over time by speakers of later languages that extrapolate the combining forms from already existing (inherited or borrowed) compounds. The further back in history we go, the better people were at playing by the ‘original’ Latin/Greek rules and keeping the two combining forms distinct; the further ahead in history we go, the more the distinction is lost, nearly always in favour of the preconsonantal form.
Thus, words like psychoanalysis are born.
Some combining forms, radio- (from Latin radius ‘beam, ray’) or pseudo- (from Greek ψεῦδος ‘lie, falsehood’) have indeed come so far that there are hardly any compounds that use the ‘correct’ form—pseudepigraphy (and derived forms) is the only one that I can think of. If they had been coined early enough, back in the Middle Ages, we would probably have radielectric, radiactive, etc. But those words were (naturally) coined much later, when radio- was simply the combining form used; and the same goes for psychoanalysis.
Some other psycho- words, like psychiatry, were actually formed in the Middle Ages, according to the ‘old’ system, and therefore have the variant combining form; while others were formed later and have generalised the preconsonantal combining form: psycho-acoustic (*psychacoustic), psycho-historic (*psychistoric), psycho-active (*psychactive), etc.
Yes, the word is “correctly constructed”, as long as you accept that the mechanisms used when constructing compounds in modern languages, based on vocabulary and combining forms from the Classical languages, are not the same as the ones that were used when the Classical languages were still in actual use and the compounds therefore were made in the language itself.