There are so many questions on this page related to time expressions let me collect them all in one comment and try to explain them. Of course, there are many others but I'll stick to those discussed here.
Αυτή η περίοδος = at this time /during this period.
"Μία φορά" = "once/one time" (not twice or more).
"αυτή τη φορά". "This time" with the meaning of "This time I managed to hit the ball, the last time I swung and missed, hope I hit it next time."
"αυτή την εποχή" = "this time of year" "εποχή" also means season ("summer" etc) or period, or age (the Age of Reason).
Hope these help.
Sorry to have missed your question. This time with the meaning of "This time I managed to hit the ball, the last time I swung and missed, hope I hit it next time." is "αυτή τη φορά".
"this time of year" is "αυτή την εποχή". "εποχή" also means season "summer" etc, or period, or age the Age of Reason.
In Greek, the concept of "time when" is usually expressed with accusative. This does not require the preposition. Θα συναντηθούμε την Δευτέρα, we will meet ON ΜΟΝDAY. However, η Δευτέρα, is simply "Monday," being nominative. You cannot translate it as "on Monday." Likewise, αυτή η περίοδος does not make sense as "at this time." Nominative case has a very strict use in Greek. It is basically used for the subject of a sentence, or a noun in apposition with the subject, or a predicate nominative (usually following a linking verb). It would logically have to be αυτήν την περίοδο, accusative, to express "at this time."
@Atheia @Joanna206443 @Walt1965 @Kalikur
This sentence is a bit... awkward. Normally, the Greek sentence should have been in accusative instead of nominative, since there is at in the Engish sentence. However, I do feel like this time would translate to αυτήν τη φορά in Greek, with the meaning of this time, and not some other time, and not to αυτήν την περίοδο, which refers to a period of time. Again, the Greek sentence and its translation is a bit weird to begin with, so I understand the confusion. :/