"The review concerns the third film."
Translation:Die Kritik betrifft den dritten Film.
Seems to me there many possibilities in German and English.
Although "betrifft" more directly translates "concerns", the gist of this sentence is more commonly expressed in English as "the review dealt with" or "the review was about" or "the review covered" or "the review had to do with".
So, FWIW, in my opinion: "Die Kritik geht um den dritten Film" and "Die Kritik handelt sich um den dritten Film" should also both be accepted.
Can "betreffen" be used reflexively? "Die Kritik betrifft sich mit dem dritten Film."
No -- that looks as if you tried to use the verb sich mit etwas befassen (to deal with (a topic), to treat (a topic)).
Die Kritik befasst sich mit dem dritten Film would be a valid German sentence: The criticism concerns itself with / deals with / concentrates on the third film.