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and be understood, but it's a question of collocation.
"Mentally ill" is by far the more popular term as the Google Ngram below confirms Collocation In Google books; "physically and mentally" yields a massive 10,200,000 results while "physically and psychologically" yields 1,340,000 results.
Either way, the users of 'psychically' meant what you mean, mentally, but possibly at a level below ordinary consciousness.
'Psychologically' can mean exactly the same thing, but it is sometimes colored with an assumption of objectivity and conscious analysis.
Romantic and ideas should be entirely banished and the steps of his investigation should be guided by intellect as cold and passionless as the instruments he uses.
Fay later confessed to her fraud and revealed the tricks she had used.
Mentally as an adverb, does not usually refer to one's sanity; to prepare oneself here the given example means to control one's nerves, anxiety, and emotional state.
You mentally challenge yourself to face the task ahead.
It's fully understandable and I'm sure many native speakers have uttered similar things in their lives, nevertheless it's hard to argue against the term, Further data Google books reports 9,960 results for "psychic problems"; 165,000 results for "mental problems" and 626,000 results for "psychological problems".
By looking at their usage, the differences between these terms will be clearer.